University  of  California  •  Berkeley 


THE 

AMERICAN   YOUTH: 


BEING 


A  NEW  AND  COiMPLETE  COURSE 

O      F 

Introductory  Mathematics  : 

DESIGNED    FOR    THE    USE    OF 

PRIVATE    STUDENTS. 


B       Y 

CONSIDER  and  JOHN   STERRY. 


VOL.     I. 


'where  the  mind 


In  endlefs  growth  and  infinite  afcent, 
Rifes  fromjiate  tofeate^  and  world  to  world. 

THOMSON, 


PRINTED  AT  PROVIDENCE, 

BY  BENNETT  WHEELER,  FOR  THE  AITTHORS, 


PREFACE. 


TIME,  ever  big  with  wonders  to  be 
unfolded  to  the  human  mind,  has  ujh- 
ered  iny  through  a  feries  of  the  mofl  important  events, 
the  rifing  Empire   of  America  \  who  hath  eftablifhed 
her  own  Independence^  and  the  flame  of  her  liberty  has 
Jpread  itjelf  to  the  remo&fl  parts  of  the  earth  j  the  cf- 
fett  of  which  great  example  has  not  yet  Jpent  its  force, 
but  mufl  continue  to  operate  throughout  ages,  and  form 
a  grand  ingredient  in  the  affive  fermentation,  and  in 
the  hiftory  of  nations* 

But  the  great  objett  of  true  national  dignity  and 
grandeur ',  conjifts  in  the  cultivation  of  the  human  mind, 
whereby  the  natural  fav age  barbarity^  rudenefs  and  im 
becility  of  human  nature  are  eradicated,  and  thofe  prin 
ciples  of  knowledge  and  virtue  engrafted  in  the  foul, 
which  are  the  foundation  of  that  knowledge  and  pre 
eminence  of  merit,  which  isthenobleft  of  all  diftinftions. 

As  foon  as  we  begin  to  exift,  that  Javage  and  im 
becile  fpirit  takes  root  in  the  Joul,  and  grows  as  the 
mind  enlarges,  tilltheje^ds  of  knowledge  by  cultivation 
do  take  effectual  root,  and  then  like  the  tender  bud 
it  will  burfl  its  native  bonds,  expand  and  flgurijh  in 
its  own  beauty  ;  Ths  veil  will  then  disappear,  and  an 

infinite 


. 
infinite  diverfity  offcenes,  both  plea 

iv  ill  open  themjelves  to  our  view.  But  in  order  to  f  re- 
fare  the  mind  for  thefe  f  leafing  and  enlarging  views, 
we  muft  early  employ  ourf elves  in  the  Jiudy  offomething 
which  is  noble  and  important,  whereby  our  minds  may 
be  cultivated  aud  brought  to  maturity.  cc  A jufl  and 
perfect  acquaintance  with  thefimple  elements  offcience, 
is  a  neceffary  Jhp  towards  our  future  progrefs  and  ad 
vancement  j  and  this,  ajfiftedby  laborious  mvejiigation, 
and  habitual  enquiry  will  conjlantly  lead  to  eminence 
and  perfection  " 

<f  But  as  the  various  modes,  fituations  and  circumftan- 
* "ces  of  life  are  various,  fo  accident,  habit  and  education, 
have  each  their  predominate  influence,  and  give  to  every 
mind  its  particular  bias"  It  is,  therefore,  for  this 
re  of  on,  we  particularly  admire  thofe  things  which  are 
the  moft  compatible  with  our  genius  and  purfuits  in  life* 

"  Riches  and  honours  are  the  gifts  of  fortune,  cafu- 
ally  beftowed,  or  hereditarily  received,  and  are  fre 
quently  abufed  by  their  foffeffbrs  -,  but  the  Juperiority  of 
ivifdom  and  knowledge,  is  a  pre-eminence  of  merit,  thai 
originates  with  the  man,  and  is  the  nobleft  of  all  dif- 
tintJions." 

Since,  therefore,  the  cultivation  of  knowledge  is  a 
thing  of  the  laft  importance,  too.many  attempts  cannot 
be  made  to  render  it  univerfal,  andjince  youth  is  the 
time  therefor,  we  have  therefore,  "  only  to  point  out  to 
them  feme  valuable  acquifition,  and  the  means  of  ob 
taining  it.  ¥be  active  principles  are  immediately  put 
in  motion,  and  the  certainty  of  the  conqueft  is  enfured 
from  a  determination  to  conquer."  But  of  all  thejdences 
cultivated  by  mankind^  none  are  more  ujeful  than  the 

Mathematics, 


Mathematics,  to  call  forth  a  Spirit  of  enterprife  and 
enquiry,  The  unbounded  variety  of  their  application^ 
which  is  of  univerfal  utility  to  mankind,  fir  ft  prompts 
our  curicfity  to  have  in  pojfeffion  a  treasure  offuch  in- 
6/limable  value.  By  their  elegant  and  fub  lime  manner 
of  reafoning,  our  minds  are  enlightened  and  our  under- 
ftanding  enlarged^  and  thereby  we  acquire  a  kabit  of 
reafoning,  an  elevation  of  thought,  that  determines  the 
mind  and  fixes  it  for  every  other  purfuit  -,  and  none  but 
thofe  who  either  from  fordid  views,  or  a  grofs  ignorance 
of  what  they  difpife,  will  ever  think  their  time  mijjpent, 
or  their  labours  ufelefs  in  the  purfuit  of  that,  which  is 
the  guide  of  our  youth,  and  the  perfection  of  our  reafon* 


of  'the  prefent  performance,  is  Arithmetic 
and  Algebra  ,  the  foundation  of  all  our  mathematical 
enquiries. 

Although  a  great  number  of  books  has  been  publijh- 
ed  on  tkefubjeff  of  Mathematics,  yet  few  of  them  are 
adapted  to  the  capacity  of  young  and  tender  minds. 
Where  is  that  fimpli  city,  plainnefs  and  brevity,  which 
is  e.bjohitely  necejj'ary  for  the  young  and  unajfijled  be 
ginner  ?  That  clofe  and  refined  reafoning  with  which  thofe 
Authors^  writings  are  replete,  renders  them  unfit  for 
learners  in  general,  and  entirely  ufelejs  to  thofe  unajjift- 
ed  by  a  'Tutor  :  They  have  ccnfulted  more  the  elegance 
of  their  diffion,  and  refined  demonft  rations,  than  the 
method  of  conveying  their  knowledge  to  their  readers. 
Others  again,  in  attempting  to  render  their  JubjetJs  at 
tainable  to  the  weakeft  minds,  have  been  fo  prolix  and 
voluminous,  as  even  to  difcourage  a  learner  at  the  fight 
of  their  works  :  Thus,  we.  fee  that  writers  in  general, 
aim  at  the  extremes,  while  the  true  and  proper  medium 
is  for  the  moft  fart  gmitsd*  Propriety  therefore,  and 

compatibility 


compatibility  ought  always  to  be  the  grand  text, 
fimpluity joined  with  brevity  leads  the  chain  of  argument. 

In  all  countries,  where  thefciences  are  cultivated^ 
local  intere/is  have  been  particularly  confidered,  which 
muft  therefore  exclude  thofe  who  neglect  the  cultivation 
if  the  Arts  and  Sciences,  from  many  advantages  of  their 
works. 

'Taking  into  ccnfideration  the  works  of  thofe  who  have 
gone  before  us  on  this  fubjefl,  the  utility  of  an  alteration 
appeared  manifeft,  while  reafon  and  convenience  urged 
the  pr -amicability  thereof. 

In  the  profecution  of  this  plan,  we  have  in  the  frft 
Book  of  the  prefent  Volume,  explained  the  rudiments 
end  application  of  numbers  ;  beginning  with  the  proper 
ties  of  an  unit,  we  have  led  the  learner  by  eafy  and  na* 
tural  gradations  to  the  moft  remote  analogies  of  the 
Jcience.  In  all  the  calculations  relating  to  money,  we 
have  made  life  of  the  Federal  Money,  or  Money  of  the 
United  .States,  which  is  not  only  much  more  ccncife  than 
the  prefent  praflice  by  pounds, /hillings,  &c.  but  it  is 
equally  eft im able  for  its  fimpli  city  and  brevity.  The 
denominations  of  this  money  being  in  a  decimal  ratio, 
are  therefore  above  all  other  numbers,  the  mvft  natural 
and  eafy  to  be  manage }d, and  which  muft  confequently  give 
it  a  preference  to  any  ofher  method  whatever. 

tfhe  futyeft  of  the  fecond  Book  is  Algebra,  or  the 
Analytic  art ;  which  above  all  others  is  the  moft  ex- 
tenfive  and  fublime.  It  was  by  this,  with  the  conji- 
deration  of  motion,  that  one  did  in  fome  meafure  do  ho 
nour  to  hnman  nature  itjelf,  by  his  almojl  divine  in 
vention*  ;  whichfucceeding  ages  will  view  with  pleaf- 
ing  admiration.  Algebra 

*  Alluding  to  Sir  Ifaac  Newton's  invention  of  Fluxions, 


C      vii      ) 

l'  Algebra  Is  a  general  method  of  dif cover  ing  truth  in 
all.  cafes  where  proper  data  can  be  ejtabiijked,  with  tbs 
greateft  expedition,  elegance  and  eafe. 

In  delivering  the  rudiments  of  thisjcience,  we  have 
particularly  confulted  the  eaje  and  accomodation  of  the 
learner y  by  confining  every  thing  within  thejphere  of  the 
ingenious  Student,  and  therefore,  exploding  thofe  tedious 
end  complicated  explanations,  which  are  commonly  to  be 
found  in  authors  on  this  JubjeR.  'The  leading  queftions 
are  Jhort  andjimple,  and  the  method  of  arguing  brief 
and  confpicuous  ;  which  particular,  although  of  the  I  aft 
importance  to  facilitate  the  progrefs  of  learners,  is  too 
much  neglected  by  mofl  writers,  and  tonfequently,  deter 
many  from  becoming  acquainted  with  this  inter  ejling  and 
important  acquirement.  Great  attention  has  been  paid 
to  render  the  doftrine  of  irrational  quantities  plain  and 
intelligible,  particularly  the  method  of  expanding  quan 
tities  into  infinite  Jeries,  and  noting  their  powers  and 
roots,  which  is  a  matter  of  the  lafl  importance  in  the 
higher  branches  of  the  Mathematics.  And  finally^ 
through  the  whole  of  the  following  Jheets,  Simplicity  and 
brevity  has  been  our  general  aim,  and  at  the  fame  time 
to  explode  all  foreign  and  provincial  cuftoms,  and  adapt 
the  whole  to  the  practice  and  convenience  of  the  United 
States. 

fhusfar,  for  the  fatisfacJion  of  the  learner,  we  have 
explained  the  economy  of  the  prefent  performance,  we 
/hail  now  fubmit  it  to  the  candid  public,  and  from  the 
pajns  we  have  taken  to  render  thefubjett  ufeful  to  learn 
ers  in  general,  are  not  without  hopes  of  its  meeting  with 
their  approbation. 

THE    AUTHORS.    ^ 

Prefton  (Connefticut)  July.,  1790. 


RECOMMENDATIONS. 


of  a  letter  from  Mr.  NATHAN  DABOLL, 
Teacher  of  Mathematics  and  Aftronomyy  in  the  Aca 
demic  School  in  Plainfisld>  to  the  Authors  -,  da  fed 
March  i,  1787. 

GENTLEMEN, 

cc  T  H  A  V  E  perufed  the  firft  Volume  of  your  new 
A  courfe  of  introdu6tory  Mathematics,  entitled 
THE  AMERICAN  YOUTH  ;  and  it  appears  to  me  a 
work  well  executed,  and  compatible  with  its  defign. 
You  have  given  your  rules  and  examples  in  a  con- 
cife,  plain  and  familiar  manner,  and  confequently 
well-adapted  your  matter  to  the  capacities  of  learn 
ers  :  I  therefore  efteem  it  a  very  valuable  perform 
ance,  and  wilh  you  fuccefs  in  its  publication,  and 
that  it  may  meet  with  an  encouragement  from  the 
public  equal  to  its  merit." 


B  Frcm 


RECOMMENDATIONS. 

From  Mr.  JARED  MANSFIELD,  to  the  Authors  \  dated 

)  December,    1787. 


GENTLEMEN, 

«  *\TO  U  R  Treatife  of  Arithmetic  and  Algebra, 
A  I  have  perufed  with  care  and  attention,  and 
have  the  pleafure  of  afluring  you  I  think  it  a  work 
of  ingenuity  and  merit.  My  reading  of  mathemati 
cal  books  hath  been  extensive  -,  yet  I  know  of  no 
writer  who  hath  treated  thefe  fubjeds  in  a  more 
fcientific  and  comprehenfive  manner,  and  at  the  fame 
time  accommodated  his  matter  fo  well  to  the  capa 
cities  of  learners,  as  I  find  to  be  done  in  your  work. 
If  you  publifh  it  (which  I  hope  you  will  not  fail  to 
do)  I  have  no  doubt  it  will  be  received  into  our 
Schools  and  Seminaries,  as  it  is  high  time  that  Ward, 
Hammond,  and  other  inferior  treatifes  now  in  com 
mon  ufe,  were  exploded.  For  my  own  part,  as  a 
lover  of  Mathematics,  I  wifh  you  all  pofiible  fuccefs* 
and  that  you  may  be  encouraged  to  proceed,  and 
write  on  the  higher  and  more  fublime  branches  of 
the  Mathematics  j  and  that  a  fpirit  of  emulation  may 
be  excited  among  the  Youth  of  America,  to  excel  in 
thefe  ufeful  and  exalted,  but  hitherto  much-neg- 
lefted  purfuits." 

>         i  ^««MiiNM^<aMBEPHBgP|PH)p|M^ 

From  • 


RECOMMENDATIONS. 

From    Col.    SAMUEL  MOTT   to   tbe  Authors,  dated 
P  reft  on,  April  28,   1788, 

GENTLEMEN, 

iC  ^rr OUR  Manufcript  Treatife  on  Arithmetic 
JL  and  Algebra,  entitled  THE  AMERICAN  YOUTH, 
has  been  put  into  my  hands.  I  have  paid  particular 
attention  in  its  perufal.  I  have  heretofore  been  con- 
fiderabiy  engaged  in  the  reading  and  ftudy  of  au 
thors  upon  the  various  branches  of  Mathematics, 
though  of  late  I  have  been  more  diverted  from  that 
purfuit.  It  has  however  given  me  great  pleafure 
and  fatisfaction  to  obferve  the  ingenuity,  concifenefs 
and  perfpicuity  which  appears  in  your  work,  noU 
withftanding  the  extenfive  and  finifhed  refearches  de^ 
monftrated  in  all  your  rules  and  examples  ;  yet  it 
appears  to  me  exceedingly  well  accommodated  to 
the  capacity  of  a  learner,  and  your  method  through 
the  whole  more  eafy  than  any  I  have  before  feen.  If 
you  fhould  publifh  your  book  (which  my  high  efteem 
fpr  mathematical  fcience,  and  fincere  regard  for  the 
progrefs  of  literature  among  the  youth  of  our  coun 
try,  induces  me  earneftly  to  wifli  you  may)  juftice 
obliges  me  to  fay,  that  I  am  clearly  of  opinion  it  will 
be  found  moreufeful  among  ftudents  than  any  other 
author  now  extant  upon  the  fubjeft.  I  fincerely  wifh 


RECOMMENDATIONS. 

you  fuccefs,  and  that  you  may  meet  with  every  en 
couragement  which  the  merit  offo  important  a  work 
deferves." 


From  the  Rev.  JOSEPH  HUNTINGTON,  D.  D.   one  of 

the  Truftees  of  Dartmouth  College  y  &c.  to  the  Hon. 
JOHN  DOUGLAS,  Efq.  dated  Coventry ,  May  23, 
1788. 

cc  T  H  A  V  E  with  much  pleafure  perufed  the  ma- 
A  thematical  competition  in  which  the  two  Mefirs. 
STERRY'S  are  united,   and  really  think  it  worthy  of 
publication    and    encouragement  :  The   fcience   of 
Arithmetic  and  Algebra  has  hitherto  been  extended 
nearly  to  its  bounds,  but  I  efteem  this  work  an  ex 
cellent  piece  for  the  iludy  of  youth,  to  lead  them  to 
the  knowledge  of  this  ufeful  fcience,  fin.ce  it  is  more 
eafy  and   intelligible  to   tender  capacities  than  any 
\vork  of  the  kind  preceding,  and  this,more  efpecially 
in  the  moft  abftrufe  part  of  the  whole  fcience,  *.  e. 
Algebra.     I  could  wifh  that  you,  Sir,  and  many  o- 
ther  gentlemen,  eminent  for  their  friendfhip  to  the 
liberal  fciences,  m'ight  pay  attention  to  the  work  I 
have  alluded  to," 


TABLE  OF  CONTENTS. 


PART      I. 

Chapter  Page 

I.  Of  Definitions  and  Illuftrations,  17 

II.  Of  Notation  or  Numeration,  20 

III.  Of  Addition  of  fimple  whole  Numbers,  22 

IV.  Of  Subtraction  of  fimple  whole  Numbers,  27 
V.  Of  Simpk  Multiplication,  30 

VI.  Of  Divifion  of  Simple  Numbers,  36 

VII.  Of  Addition  of  Compound  Quantities  48 

VIII.  Of  Subtraction  of  Compounds,  60 

IX>  Of  Multiplication,  &;c.  of  Compounds,  63 

X.  Of  Reduction,  6S 

PART     II. 

I.  Of  Definitions  and  Illuftrations,  J6 

II.  Of  Redudion  of  Vulgar  Fra&ions,  78 

III.  Of  Addition,  &c.  of  Vulgar  Fractions,  95 

P  "A  R  T     III. 

I.  Of  Definitions  and  Illuftrations,  100 

II.  Of  Addition,  &c.  of  Decimal  Fractions,  102 

III.  Of  Redu&ion  of  Decimals,  114 


A  Supplement  to  P  A  R  T    III. 

Chap.  Page 

I.  Of  Definitions  and  Illuftrations,  119 

II.  Of  Reduction  of  circulating ;  Decimals,       121 

III.  Of  Addition,  &c.  of  circulating  Decimals,  127 

A  Supplement  to    PART    I. 

I.  Of  Proportion,  or  Analogy,  132 

II.  Of  Disjunct  Proportion,  149 

III.  Of  Simple  Intereft,  162 

IV.  Of  Compound  Intereftj  179 
V.  Of  Rebate,  or  Difcount,  183 

VI.  Of  Equation  of  Pay ments,  187 

VII.  Of  Barter,  189 

Vill.  Of  Loft  and  Gain,  191 

IX.  OfFellowfhip,  193 

X.  Of  Compound  Proportion,  200 

XL  Of  Conjoined  Proportion,  204 

XII.  Of  Allegation,  206 

XIII.  Of  Pofnion,  or  the  Gueffing  Rule,  214 

XIV.  Concerning  Permutation  &  Combination,  218 
XV.  Of  Involution,  228 

XVI.  Of  Evolution.,  229 

BOOK    II. 

I.  Of  Definitions  and  Illuftrations3  241 

II.  Of  Addition  of  whole  Quantities,  245 

III.  Of  Subtraction  of  whole  Quantities,  248 

IV.  Of  Multiplication,  249 

V.  OfDivifion,  252 
VI.  Of  In  volution  of  whole  Quantities,  257 

VIL 


Chap.  f  Page 

VII.  Of  Multiplication  and  Divifion  of  Powers,  265 

VIII.  Of  Evolution  of  whole  Quantities,  268 

IX.  Of  Algebraic  Fradtions,  273 

X.  Concerning  Surd  Quantities,  285 

XI.  Of  infinite  Series,  296 

XII.  Of  Proportion,  306 

XIII.  Of  fimple  Equations,  315 

XIV.  -Of  Extermination  of  unknownQuantities,  321 
XV.  Solution  of  a  Variety  of  Queftions,  330 

XVI.  Of  Quadratic  Equations,  341 

XVII.  Solution  of  Queftions  with  Quadratics,    348 

XVIII.  Of  the  Generis  of  Equations,  355 

XIX.  Of  the  Transformation  of  Equations,  &c.  362 
XX.  Of  theRefolution  of  Equations  byDivifors,  368 

XXI.  Of  finding  the  Roots  of  numeral  Equa 

tions,  by  the  Method  of  Approximation,  373 

XXII,  Concerning  unlimited  Problems,  377 


ALTHOUGH  the  Authors  ex 
amined  the  Proof-Sheets^  yet  the  follow 
ing  efcaped  their  Notice. 

ERRATA. 

PAGE  28,  laft  line,  dele  See  the  Example.  P. 
34,  1>  12.  read  69530000.  1.  14,  r.  720800.  P.  35, 
1.  1 8,  for  3,  r.  4.  P.  55, 1.  4,  f.  content.  1.  24,  for 
3  qr.  3  na.  r.  i  qr.  3  oa.  1.  26,  for  3qr.  2  na.  r.  i. 
qr.  2  na.  P.  67,  1.  4,  r.  105  dol.  P.  74,  1.  9,  r. 
5638^8.  1.  19,  r.  15480  yards.  P.  77,  L  20,  for  in, 
r.  is.  P.  80,  1.  13,  for  13,  r.  15.  P.  85,  1.  i,  for 
2o»%-  -;-  20,  r.  20-0  —24.  P.  ioo,  1.  15,  for  .5,  r. 
5.  P.  1 06,  L  20,  r.  preceding.  P.  107,  1.  27,  r. 
8rr.  P.  113,  1.  12,  r.  31.415*  &c-  p-  i32>l-25, 
r.  numbers.  P.  157,  1.  13,  r.  operation.  P.  169, 
I.  13,  r.  49cts.  P.  193,  I.  7,  r.  51  dol.  72^4  cts. 
1.  15,  r.  fcllowfhip.  P.  322,  1.  5,  r.  6X2X6.  P. 
245, 1.  16,  for  -j-20,  r.  +2#.  P.  246,  for  ax>  r.  az. 
P.  249,  for  \/aw  — yby  r.  ^aw — yb.  P.  266,  1. 

i6,r.  i+j^-5-jf-f.yl*.  P.  288,  for  a*,  r.  a*.  P. 
353,  1.  1 6,  for  laft,  r.xvi.  P.  357,!.  19,  r.  v  =  — 
c-  IJ-  379?  1-  9>  ^ad  axiom  8.  J.  10,  r.  axiom  8, 
1.  1 8,  r.  axiom  8.  P.  380,  1.  i,  r.  ax.  7.  1.  2,  r. 
ax.  9.  1.  3,  r.  ax.  8.  1.  5,  r.  ax.  8. 


BOOK       I. 

OF    ARITHMETIC. 


!>OO<>^^ 

PART    I. 

4RITHMETIC  of  WHOLE  NUMBERS. 

CHAP.     L 
Of  DEFINITIONS  and  ILLUSTRATIONS. 

ARITHMETIC  confift&of  three  parts';  twa 
of  which  are  natural,  and  the  third  artificial* 
The  firft  part  of  natural  Arithmetic,-  is  wherein  an 
unit  or  integer  reprefents  one  whole  quantity,  of  any 
kind  or  fpecies  ;  and  is  therefore  ftiled  Arithmetic 
of  whole  numbers.  The  fecond  part  of  natural  A- 
rithmetic,  is  wherein  an  unit  is  confidered  as  broken 
or  divided  into  parts,  either  even  or  uneven,  which 
are  considered  either  as  pure  parts  of  an  unit,  or  as 
parts  mixed  with  an  unit  -,  and  is  ufually  ftiled  the 
doctrine  of  vulgar  fractions .  The  third  parr,  or  ar 
tificial  Arithmetic,  is  an-  eafy  and  elegant  method  of 
managing  fractional,  or  broken  quantities  ;  the  oper 
ations  are  nearly  fimilar  to  thofe  of  whole  numbers* 
This  part  is  of  general  ufe  in  the  various  branches  of 
the  Mathematics. 

C  THE 


(       18        ) 

THE  operations  of  common  Arithmetic  in  all  its 
parts,  are  performed  by  the  various  ordering  and 
difpofmg  of  ten  Arabic  characters,  or  numeral  fi 
gures  ;  which  are  thefe  following,  viz. 
one  two  three  four  five  fix  feven  eight  nine  cypher 
1234567  8.  9  ° 

AN  unit  (by  Euclid)  is  that  by  which  every  thing 
that  is,  is  one  ;  and  number  is  compofed  of  a  multi 
tude  of  units* 

NINE  of  the  aforefaid  figures,  are  compofed  of 
units  ;  each  character  reprefenting  fo  many  units  put 
together  in  one  fum,  as  was  intended  they  fhould  de 
note  ;  nine  of  thofe  units,  being  the  greateft  number 
which  is  thought  bed  for  any  one  character  to  re- 
prefent  ;  the  lail  of  the  before-mentioned  characters, 
is  a  cypher,  or  as  fome  call  it  a  nothing  ;  for  of  itfelf 
it  is  nothing  ;  becaufe,  if  ever  fo  many  cyphers  be 
added  to,  or  fubtracted  from  an  unit  or  number, 
they  will  neither  increafe  nor  diminifh  its  value  :  con- 
fequently  a  cypher  of  itfelf  is  no  aflignable  quantity  ; 
but  cyphers  annexed  or  prefixed  to  an  unit  or  num 
ber,  will  increafe,  or  diminifh  that  unit  or  number 
in  a  tenfold  proportion. 

THAT  the  learner   may  underftand  the  following 
fheets,  it  is  abfolutely  necefifary  for  him  to  be  well 
acquainted  with  the  following  Algebraic  figns. 
SIGNS  &  NAMSS.  SIGNIFICATIONS, 

fis  the  fign  of  Addition  :  as  4+6, 

+  Plus,  or  more,]  w,fich  d™°<-es.that  6  is  to  be  add- 
M  ed  to  4,  and  is  read  thus,—  • 
(.4  more  6. 
is  the  fign  of   Subtraction:    as 

4~~2>  which        nifies  that  2  is  to 


X  into, 


Minus  o 
us,o 

.thus,  4  lefs  2. 


X  into,  or  with, 


by, 


~  equal, 


fo  is, 


is  the  fign  of  Multiplication:  thus 
4X3  denotes,  that  3  is  to  be  mul 
tiplied  into  4  ;  and  is  read  thus, 
4  into,  or  with,  3. 
is  the  fign  of  Divifion  :  thus 
6-:-3,  is  6  divided  by  3,  or  f,  fig- 
nifies  the  fame  thing  ;  and  is  read 
thus,  6  by  3. 

is  the  fign  of  Equality  :  and 
whenever  this  fign  is  placed  be 
tween  any  two  quantities,  it  de 
notes  that  thofe  quantities  are 
equal  :  thus  9—9  ;  that  is,  9  e- 
quals  9  ;  alfo  6+4—10,  is  read 

C6  more  4  equals  10. 

pis  the  fign  of  Proportionality  ; 
and  is  always  placed  between  the 
iecond  and  third  numbers  that 
are  in  proportion  :  thus 


ii 

p  is  alfo  a  fign  of  Proportion,  and  is 

I  placed  between  the  firft  and  fe- 
:  to,  \  cond,  third  and   fourth  numbers 

i  in  proportion  :  thus  a:4::j:6; 

is  read  thus,  2  to  4  fo  is  3  to  6. 
A  +6X2  ("denotes  the  fum  of  4  &  6  mul- 

Jtiplied  with  2. 
fr  is  the  fign  of  continued  Proportion. 

THE  whole  doftrine  of  Number  is  founded  on 
the  five  following  general  rules,  to  wit,  Notation, 
Addition,  Subtraction,  Multiplication  and  Divifion, 


CHAP.     IL 


CHAR     IL 


O/  NOVATION  or  NUMERATION. 

TV  T  OTATION  or  Numeration    teaches    us, 
J[\|    how  to  exprefs  the  value  of  figures  ;  and  con- 
iequently  to  note  or  write  do\jn  any  propofed  num— 
'l}er,  according  to  itsjuft  value  ;  in  the  operation  of 
which,  two  things  muft  be  obferved,  viz,  the  order 
of  writing  down  figures,  and  the  method  of  valuing 
each  in  its  proper  place,  as  in  the  following  Table  ; 


:ON  TABLE. 

nds  of  Millions 

(A 

a, 

0 

1 

o 

</» 

<Si 

to 

C 

*"«^ 

f% 

CO 

/"» 

n 

••n 

NUMERAT 

Hundreds  of  Thouf 

Tens  of  Thoufand 

Thoufands  of  Millie 

Hundreds  of  Millio 

en 

G 

.2 

i 

0 

s 

H 

Millions 

Hundreds  of  Thouf 

Tens  of  Thoufands 

Thoufands 

Hundreds 

C/3 

S  [5 

3 

2 

i 

9 

8 

7 

6 

5 

4 

3 

2    I 

HERE, 


HERE  the  order  of  reckoning  begins  on  the  right 
hand,  to  wit,  at  unity,  and  foon  as  the  table  directs. 
But  to  make  the  underftanding  of  this  table  plain, 
it  is  required  to  exprefs  the  value  of  the  numeral  fi 
gures  321.   Firft,  beginning  at  the  firft  figure  on  the 
right  hand,  viz.  at  i,  which  ftands  in  the  units*  place, 
where  it  reprefents  its  own  fimple  value,  which  is  an 
unit,  or  i  ;  the  next  tc^be  confidered  is  the  figure  2, 
which  ftands  in  the  tens1  place,  reprefenting  lo  many 
tens,  as  the  figure  2  is  compofed  of  units,  which  are 
two ;  fo  that  the  figure  2  Handing  in  the  place  of 
tens  reprefents  2   tens,  or  20  ;   the  next  figure,  3, 
{lands  in  the  hundreds'  place,,  and  fignifies  as  many 
hundreds  as  the  laid  figure  hath  units,  viz.  3,  that  is, 
three  hundred:  now,  if  the  whole  value  of  thejigures 
321  beexpreiTed,  the  expreflion  will  be  three  hundred 
twenty-one.     Altho  the  figure  3,  is  in  the  laft  place 
on  the  right,   or  the   firft   on  the  left,    yet   when 
we  come  to  read  or  exprefs  them,  we  begin  with  the 
figure  3  ;  becaufe  the  method  of  reading  figures  is 
the  fame  as   that   of  words.     Hence   the  firft  fi 
gure  in    numbering,    is  the  firft  figure  on  the  right 
hand  ;  but  in  reading  or   exprefiing   the   value  of 
numbers,    the  firft   figure  in  the  exprellion    is  the 
firft   figure   on  the  left   hand.     Again,  let  it  be  re 
quired  to  read  or  exprefs   7645.     Here  as  before, 
the  firft  figure   of  the  propofed  number,  to  wit.  5, 
ftands  in  the  units'  place,  and  is  5  units,  or  five,   the 
fecond  figure  which  is  4,  is  in  the  tens'  place,  and  is 
four  tens  or  40,  the  third   figure  which  is  6,  in  the 
hundreds'  place,  is  called  hundreds,   and  the  fourth 
figure,  which  ftands  in  the  thoufands'  place,   is  for 
the  fame  reafori  called  thoufands  ;  and  the  expreflion 
for  the  whole  value,   beginning  as  before,   is  feven 
thoufand  fix  hundred  forty-five. 

IF 


IF  what  has  been  laid  concerning  notation  and  va 
luation  of  figures,  be  thoroughly  confidered,  toge 
ther  with  the  following  examples  and  their  anfwers, 
the  whole  bufmefs  of  Numeration  will  appear  plain 
to  the  meaneft  capacity* 

EXAMPLES. 

What  is  the  value  of  56434  ? 
Anfwer,  Fifty -Jix  thoufandfour  hundred  thirty -four. 
What  is  the  value  of  7843217  ? 
Anf.     Seven  million  eight  hundred  forty-three  thou- 
Jand  two  hundred ftvente  en. 

What  is  the  value  of  640036  ? 

Anf.  Six  hundred  forty  thoujand  thirty -fix. 

What  is  891000002  ? 

Anf.  Eight  hundred  ninety -me  million  two. 


CHAP.     III. 

Of  ADDITION  of  SIMPLE    WHOLE  NUM 
BERS'. 

ADDITION  is  the  collecting  or  putting  to 
gether  feveral  quantities  or  numbers  into  one 
fum,  fo  that  their  total  amount  may  be  known  :  and 
in  order  to  perform  the  operations  of  this  rule,  two 
things  muft  be  carefully  obferved,  which  are,  Firft, 
the  right  placing  or  fetting  each  figure  in  its  proper 
place  j  that  is,  units  muft  Hand  under  units,  tensun- 
Uer  tens,  hundreds  under  hundreds,  andfoon,  fetting 
each  denomination  under  that  of  the  fame  value: 
thus  246  +  25  +  163,  being  fee  asdireded,  will  ftand 

thus, 


f246 

,  <    25 
(163 


thus,  <    25 
63 

THE  fecond  thing  to  be  ohferved,  is  the  right  col 
lecting  or  adding  together  each  perpendicular  row 
of  figures,  placed  as  before  directed  •,  which  is  per 
formed  as  in  the  following  example,,  being  the  fame 
as  made  ufe  of  above,  viz.  246  +  25-1-163  : 


f 
< 

(. 


or  thus,    <{    25 
163 

Then  ftriking  a  line  beneath  the  figures,  as  in  the 
example  ;  begin  on  the  right  hand  at  the  units'  place, 
adding  together  all  thofe  figures  which  (land  in  the 
units'  place,  and  if  their  fum  be  under  ten,  fet  it  down 
underneath  in  the  units1  place  •,  but  if  their  fum  ex 
ceed  ten,  fet  down  the  furplus,  carrying  one  to  the 
next  place,  viz.  the  tens*  place  :  or,  more  generally, 
as  many  tens  as  the  fum  of  thofe  units  amounts  to, 
you  muft  carry  to  the  next  place  of  figures,  to  wit, 
the  tens'  place,  adding  them  up  with  all  the  figures 
that  (land  in  that  perpendicular  line  -,  and  fo  on  for 
the  reft  ;  remembering  to  carry  one  for  every  ten  of 
your  aggregate  :  the  whole  of  which  will  be  illuf- 
trated  in  the  following 

EXAMPLE. ; 

Find  the  fum  of  the  following  numbers,  viz* 
392+466  +  256. 

THOSE  numbers  being  placed  as  the  rule  directs, 
will  (land 


thus,!  466 

se 

1 1  i4=fum  required.  Then 


Then  begin  with  the  bottom  figure,  in  the  units* 
place  5  laying  6  and  6  is  12,  and  2  is  14;  fetting 
dov/n  4,  carry  i  to  the  next,  or  place  of  tens,  faying 
5  and  i  that  I  carry  make  6,  and  6  is  12,  and  9  is 
21  j  here  becaufe  the  aggregate  or  fum  total  is  21 
units  (or  becaufe  it  (lands  in  the  tens'  place)  2  tens 
and  one  unit  ;  therefore  fet  down  i  and  carry  2  to  the 
next  place,  faying  2  and  2  that  I  carry  make  4,  and 
4  is  8,  and  3  is  1 1  j  which  being  the  ium  of  the  laft 
place  of  figures  in  the  example,  fee  down  the  whole. 
[See  the  work  ac  the  bottom  of  the  preceding  page.] 
THE  reafon  of  fetting  down  the  furplus,  or  odd 
figures,  and  carrying  for  the  tens,  as  in  the  laft  and 
all  other  examples  in  addition  of  fimple  quantities, 
is  to  Ihorten  the  work  under  coniideration  j  and  to 
faye  the  trouble  of  ufmg  fuperfluous  figures.  To 
exemplify  which,  let  us  make  ufe  of  the  foregoing 
example,  to  wit,  3.92+4664*256,  which  muft  be 
placed 


thus, 


3 

4 

2 

9 

I 

2 

6 
6 

2 

9 

i 

0 
0 

4 
0 
0 

the  fum  of  the  row  of  units 
thejum  of  the  row  of  tens 

the  fum  cf  the  row  of  hundreds 

i 

i 

4  j  the  J  urn  of  the  whole  ; 

then  adding  up  each  fingle  row,  frt  down  its  fum  in 
its  proper  place,  in  the  fame  manner  as  if  there  were 
but  one  fingle  row  ;  fupplying  the  vacant  places  on 
the  right  hand  with  cyphers.  Hence  the  refult  of 
this  operation  is  the  fame  as  in  the  former  method  of 
carrying  for  the  tens  j  and  hence  alfo  it  appears,  trlat, 
adding  the  cyphers,  makes  no  alteration  in  the  value 
of  the  fum  of 'the  other  figures. 

THE 


THE  manner  of  proving  your  work.,  flows  as  a  na 
tural  confequent,  from  the  following  felf-evidenc 
proportion,  on  which  the  truth  of  the  rule  depends, 
viz.  that  every  whole  is  equal  to  all  its  parts  taken 
together.  Wherefore  if  you  divide,  or  feperate  the 
given  numbers  into  two,  or  more  parcels,  according 
to  your  propofition  ;  and  by  adding  together  each. 
part  fo  feperated,  if  the  fum  of  all  thofe  pans  added 
together,  is  equal  to  the  fum  total  of  all  the  given 
numbers,  found  before  Operation,  your  work  is  right. 
THIS  method  will  appear  plain  by  the  following 
example.  Suppofe  it  were  required  to  add  together 
the  following  numbers,  viz.  3489  +  6725  +  2324+ 
6744  ;  which  according  to  the  rule  of  Notation  muft 
ftand  thus,  34^9 

6725 

2324 

6744 


Firftpartj 


19282  —fum  before  federation. 

Second  C  2324 
part     [6744 


firft  fart. 


z:/#tf2  of 
fecond  fart. 


The  fum  oftbefrfi  and  fecond  parts  | 


Sum  of  all  the  farts    19282 

which  agrees  with  the  fum  total  before  feperation  ; 
therefore  the  work  is  right.  But  themoft  ufual  me 
thods  of  proving  Addition,  is  either  by  beginning  at 
the  top,  and  reckoning  downwards  ;  which  fum,  if 
equal  to  that  found  by  cafting  upwards,  the  work  is 
right.  Or,  firft  add  together  all  the  propofed  num- 
D  bers 


(         26         ) 

hers  into  one  fum  ;  then  feperate  the  upper  number 
from  the  reft,  by  a  line,  and  add  together  the  re 
maining  numbers  beneath;  placing  their  fum  under 
the  former,  or  fum  total  before  feperation;  which  be 
ing  done,  add  the  fum  lad  found  to  the  upper  line  in 
your  example;  which  fum,  if  equal  to  the  fum  total 
or  firft  addition,  the  work  is  right :  this  is  the  fame 
in  effect,  as  the  firft  method  of  proof,  though  a  little 
different  in  mode,  as  will  appear  by  the  following 
example. 

34673 

24532 
12760 
53865 
21671 

1 47  506  —fum  of  the  whole 

1 1 2$ 2%— fum  of  all  but  the  upper  line 

147506=34678  + 1  i<i%i%~fum  oftbe  wMe  : 

therefore  the  work  is  right. 

TAKE  the  following  examples,  without  their  an- 

fwers,  for  practice. 

a  6538764 

3457643  460039  372  875623 
4567012  914321  42734  43521 
2354123  675422  8173456  6300 
1678432  342310  37240  579 
,  421  $4 


CHAR 


CHAP.     IV. 

Of  SUBTRACTION  of  SIMPLE  WHOLE  NUM 
BERS. 

SUBTRACTION  is  the  taking  one  number 
out  of  another,  whereby  the  remainder,  differ 
ence,  or  excefs  may  be  known  :  thus  3  taken  out  or 
from  5,  leaves  2,  which  is  the  difference  between  3 
and  5  ;  and  is  alfo  the  excefs  of  5  above  3. 

HENCE  it  follows,  that  the  number  from  which 
fubtraftion  is  to  be  made,  muft  be  equal  to,  or 
greater  than  the  fubtrahend,  or  number  to  be  fub- 
tra&ed  ;  and  alfo,  that  Subtraction  isthereverfe  of 
Addition  ;  for  Subtraction  is  the  taking  of  one  num 
ber  from  another,  but  Addition  is  the  collecting  or 
putting  them  together. 

HERE  the  Notation  is  the  fame  as  in  Addition, 
to  wit,  thofe  numbers  which  are  of  like  value,  muft 
ftand  directly  beneath  each  other ;  that  is,  units  muft 
ftand  under  units,  tens  under  tens,  &c.  After  having 
thus  placed  your  numbers,  the  lefs  beneath  the  great 
er,  you  may  proceed  to  fubtract  them  apart,  by  ob- 
ferving  the  following 

RULE. 

BEGIN  with  the  firft  figureon  the  right-hand,  which 
(lands  in  the  units'  place,  and  fubtract  the  lower  fi 
gure  from  that  which  (lands  directly  over  it,  of  the 
fame  value ;  fetting  down  the  remainder  (if  any)  be 
neath  in  the  units'  place  :  If  the  figure  in  your  fub 
trahend  be  equal  to  the  figure  which  (lands  directly 
over  it,  you  muft  fet  a  cypher  for  the  remainder ; 
but  if  the  lower,  or  figure  in  your  fubtrahend,  con 
tains  more  units  than  your  upper  figure,  you  muft 
add  x.o  to  the  upper  figure,  or  fuppofe  it  to  be  fo  add 
ed 


ed  in  your  mind  ;  then  fubtrac~b  your  lower  figure 
from  your  upper  fo  increafed,  letting  down  the  re 
mainder  or  difference  in  its  proper  place  ;  then  pro 
ceed  to  your  next  place  of  figures  -,  now  it  is  fuppof- 
cd  that  the  10  you  before  added  was  borrowed  from 
your  next  fuperior  place  of  figures,  where  you  muft 
pay  what  you  before  borrowed,  which  is  performed 
as  the  uiual  method  is,  by  calling  the  lower  figure, 
{landing  in  that  place,  one  more  than  it  really  is  ; 
then  fubtrafting  it  fo  augmented,  from  your  upper 
figure,  or  figure  ftanding  direftly  over  it,  fet  down 
the  difference  as  before  dire&ec};  and  foon,  from  one 
place  of  figures  to  another,  until  the  whole  be  complet 
ed  j  the  whole  of  which,  is  illuftrated  in  the  following 

EXAMPLES. 

SUPPOSE,  that"  from  4567,  you  were  to  fubtraft 
3692  ;  which  numbers,  being  placed  according  to 
the  rule,  will  (land 


HERTS  begin  with  the  2,  faying  2  from  7  and  there 
remains  5,  fetting  it  down  as  directed  -,  then  proceed 
to  your  next  place  of  figures,  faying  9  from  6  I  can 
not,  becaufe  my  lower  figure,  to  wit,  9,  contains 
more  units  than  my  upper,  or  figure  from  which  I 
would  fubtract;  therefore  I  fuppofe  10  to  be  added  to 
the  upper  figure  which  makes  16  j  then  faying  9  from 
1  6  and  there  remains  7  3  then  proceed  to  the  next 
place,  where  you  muft  pay  what  you  have  borrowed, 
by  faying  6  and  i  that  I  borrowed  make  7  ;  then  7 
from  5  I  cannot,  but  7  from  54-10—15,  and  there 
remains  8  ;  then  to  the  next  place,  faying  3  and  i  that 
I  borrowed  make  4,  4  from  5  and  there  remains 
i  >  now  there  being  no  more  places  of  figures,  fet 
down  the  i  and  the  work  is  done.  (See«the  eisamplo?) 

THE  ' 


(          29          ) 

THE  truth  of  fubtraction  is  founded  on  the  fame 
felf-evident  proportion,  or  axiom,  as  that  of  Addition) 
viz.  the  whole  is  equal  to  ajl  its  parts  taken  together. 
From  which  propofition  is  deduced  the  following 
method  of  proving  your  work,  to  wit,  by  adding  the 
fubtrahend,  or  number  to  be  fubtradted,  to  the  re 
mainder  :  for  the  number  from  which  fubtraclion 
is  made,  is  here  confidered  as  the  whole,  and  the  fub 
trahend,  as  a  part  of  that  whole ;  coniequently  if  that 
part  be  taken  from  the  whole,  the  remainder  will  be 
the  other  part ;  therefore  if  both  parts  when  added 
together,  be  equal  to  the  whole,  the  work  is  right. 

HENCE  it  is  manifeft  that  fubtraction  may  be 
proved  by  fubtraction  ;  for  if  from 

67834  the  whole, 
is  taken  53723  a  part  of  that  whole, 

there  will  remain,  14111  the  other  part ; 
and  if  from  67834  the  whole,  there  is  taken 
the  laft  part  14111 

there  will  remain  53723  the  firil  part,  or  fubtrahend: 
confequently,  &c. 

AGAIN,  if  from  17942  the  whole, 

is  taken  13724  a  part  of  that  whole  $ 

there  will  remain  14218  the  other  part, 

27g42~fum   of  the    fubtrahend 
and  remainder— the  whole. 

TAKE  the  following  examples  for  practice. 
From    37654       394076       2876955       7654i09 
take      28765        123468         423610         347472 

Rem.' 


CHAP.     V. 


M 


(       3Q       ) 

CHAP.     V. 

Of    SIMPLE    MULTIPLICATION. 

ULT  I  PLICATION  is  a  rule  by  which 
a  given  number  may  be  increafed  any  num 
ber  of  times  propofed. 

THERE  are  three  requifites  in  Multiplication  :  firft, 
the  multiplicand,  or  number  to  be  multiplied  :  fec- 
oncl,  the  multiplier,  which  denotes  how  many  times 
the  multiplicand  is  to  be  taken  ;  for  by  Euclid^  as 
many  units  as  there  are  in  the  multiplier,  fo  many 
times  is  the  multiplicand  to  be  added  to  itfelf :  third, 
the  producl,  or  multiplicand  increafed  fo  many  times 
as  there  are  units  in  the  multiplier. 

SUPPOSE  for  example,  that  7  be  increafed  4  times ; 
that  is,  to  multiply  7  into  or  with  4 ;  thefe  numbers 
muft  be  placed  as  in  Addition, 
,         T    7  multifile  and 
JS>  1   4  multiplier 

28  produft. 

Now  that  4  times  7  make  28,  will  appear  evident 
by  fetting  down  the  multiplicand  4  times,  and  adding 
up  the  whole,  as  in  this,  f"  7 

7 
7 
7 

i%~fum  or  fro  du  ft. 

HENCE  it  is  plain,  that  multiplication  is  a  con- 
cife  method  of  Addition. 

BUT  before  you  proceed  any  further  on  the  fub- 
jeft  of  multiplication,  you  muft  learn  the  following 

Table  : 

MUL- 


MULTIPLICATION  TABLE. 


I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

2 

4 

6 

8 

10 

12 

H 

16 

18 

20 

22 

24 

3 

6 

9 

12 

'5 

18 

21 

24 

27 

3° 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

4° 

44 

48 

5 

10 

IS 

20 

25 

3° 

35 

40 

45 

5° 

55 

60 

6 

12 

18 

24 

3° 

36 

42 

48 

54 

60 

66 

72 

7 

H 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

1  08 

10 

20 

3° 

40 

5° 

60 

70 

80 

90 

IOO 

IIO 

120 

ii 

22 

33 

44 

55 

66 

77 

88 

99 

I  IO 

121 

I32 

12 

24 

36 

48 

60 

72 

84 

96 

1  08 

120 

132 

144 

FOR  an  explanation  of  the  foregoing  Table,  fup- 
pofe  that  it  were  required  to  find  the  product  of  3X4. 
Firft,  look  in  the  left  hand  column  for  3,  and  right 
oppofite  with  it  in  the  column  under  4  at  the  top,  is 
12,  the  product  of  3X4. 

AGAIN,  to  find  the  product  of  9X12.  Look  for 
9  in  the  left  hand  column  as  before,  and  right  oppo- 
fite  to  it,  under  12  in  the  upper  column,  is  108,  the 
product  required ;  and  the  like  is  to  be  underftood 
of  all  the  reft. 

HAVING  given  you  this  fhort,  but  comprehenfive 
idea  of  the  foregoing  Table,  we  lhall  now  proceed  ta 

examples, 


(  _£_> 

examples,  with  this  caution^  to  wit,  that  in  multiply 
ing,  care  muft  be  taken,  that  the  product  of  the  firil 
figures,  ftand  directly  under  its  multiplier ;  alfo  re 
membering  to  carry  i  for  every  10  of  the  product. 

EXAMPLES. 

IT  is  required  to  multiply  120X945  which  placed 
as  before  directed  will  ftand  thus, 

1 20  multiplicand 
94  multiplier 


••--. 


HERE  you  begin  with  that  figure  of  your  multi 
plier,  which  (lands  in  the  units*  place,  viz.  4,  faying 
4  times  o  is  o,  which  fet  down  directly  under  the 
figure  you  are  multiplying  with ;  then  fay  4  times  2 
is  8,  which  fet  under  the  9;  then  4  times  i  is  4,  which 
alfo  place  as  in  the  example ;  and  the  product  of  the 
multiplicand  with  the  firft  figure  of  your  multiplier, 
is  480  :  then  begin  with  the  next  figure  of  your  mul 
tiplier,  faying  9  times  o  is  o,  which  place  under  your 
multiplying  figure,  then  fay  9  times  2  is  185  here  fee 
down  8  and  carry  i  to  the  next  place,  faying  9  times 
i  is  9,  and  i  that  I  carry  makes  10;  now  this  being 
the  product  of  the  laft  place  of  figures,  fet  down  the 
whole,  and  the  product  of  the  multiplicand,  with 
the  fecond  figure  of  your  multiplier  is  1080,  or  more 
properly  10800  :  then  adding  up  both  products, 
their  fum  is  11280,  the  product  required.  (See  the 
example  above.) 

IT  is  required  to  multiply  2439X421  -9  thefe  num 
bers  placed  as  directed  will  ftand 

thus, 


(      33      ) 

thus,  -j    4™  i  faffors 

0/2439X1 
«/  2439X400 
0/2439X421 


THE  annexing  of  cyphers,  as  in  the  lad  example, 
is  to  fupply  the  vacant  places  ;  and  to  fhew  the  fev- 
eral  products  are  fncreafed  in  a  tenfold  proportion, 
with  regard  to  the  places  in  which  your  multiplying 
figures  (land.  Thus  the  product  of  the  multiplicand 
with  the  fecond  figure  of  your  multiplier,  is  not  the 
product  of  2439X2,  but  the  produ6l  of  2439X2  tens 
or  20  ;  which  product  is  10  times  more  than  it  would 
have  been,  had  the  multiplying  figure  (2)  flood  in 
the  units'  place  -,  fo  alfo  the  annexing  of  two  cyphers, 
as  in  the  product  of  the  multiplicand  with  the  third 
figure  of  the  multiplier,  to  wit,  4,  is  becaufe  that 
figure  Hands  in  the  hundreds'  place;  and  therefore 
the  product  is  not  2439X4*  but  really  the  product 
of  2439X400  j  yet  thofe  cyphers  may  be  omitted, 
by  obferving  the  direction  in  the  beginning  of  this 
chapter,  viz.  that  the  firll  figure  of  the  feveral  pro 
ducts  (land  directly  beneath  its  correfponding  figure 
®f  the  multiplier. 

Find  the  product  of  24354X  32001 


24354 
48708 
73062 

779352354=  24354X32001  ~^mfo#  required, 
E  "  HERE 


(    J4 ) 

HERE  you  may  obferve  that  we  pafs  the  cyphers, 
taking  care  only  to  place  the  next  figure  according 
to  the  foregoing  directions. 

WHEN  there  are  cyphers  on  the  right-hand  of  the 
multiplicand,  or  multiplier,  or  to  both,  you  may  mul 
tiply  the  figures  as  before,  neglecting  the  cyphers, 
until  you  have  found  the  product  of  the  digets  only  ; 
to  which  annex. fo  many  cyphers  as  there  are  in  both 
faftors :  as  in  thefe, 


848                             6953000fic34765°X  200 
636 

720820^:21200X34 

•MB 

24000000?    ,  ' 

24000000 \J 


96 

48 


57600000000000011 24000000  X  24000000 


IF  it  be  required  to  multiply  any  number  with 
10,100,1060,  &c.  you  need  only  annex  to  your  mul 
tiplicand  fo  many  cyphers  as  are  in  the  multiplier, 
and  the  work  is  done  ;  as  in  the  following, 

20X100-2000 
300X1000=300000 
26460X1000012:264600000 

HERE  it  may  perhaps  be  ufeful,  to  acquaint  the 
learner  of  the  method  of  performing  Multiplication 
by  Addition  -9  which  in  fome  cafes  will  be  found  ufe 
ful  : 


ful :  the  method  is  as  follows  :  firft,  fet  down  the  9 
digets,  or  numeral  figures,  in  a  fmall  column  made  for 
that  purpofe  -,  then  againft  i,  place  the  multiplicand, 
againft  2,  double  the  multiplicand,  againft  3,  three 
times  the  multiplicand,  and  fo  on  to  the  laft. 

Find  the  product  of  2439X421  by  Addition, 


do. 
do. 
do. 
do. 
do. 
do. 
do. 
do. 


i 

2439irtf/fl 

'I  tip  I 

2 

3 

4878=2  times 
7317:1:3   do. 

4 

9756=4 

do. 

5 

12195=5 

do. 

6 

14634=6 

do. 

7 

17073=7 

do. 

8 

19512=8 

do. 

9 

21951=9 

do. 

r  i  is     2439 

. 

againft  <  2        4878 

Sum 


z:pro.  req. 


HERE  it  is  evident,  that  the  foregoing  table  will 
ferve  let  the  multiplier  be  any  number  whatever  ; 
for  fuppofe  it  were  required  to  find  the  product  of 
2439><6734- 

OPERATION. 


againft 


9756=2439X4 

73^7  —2439X30 

17073    —2439X700 

14634      1^2439X6000 


Sum 


EXAM- 


EXAMPLES. 

691861X26=17988386 

346732X6523:226069264 
7  9° 1 37  5X3000011237041 250000 
129186X981:12660228 
7600 1  >£  1 302:1:98953302 
3581X2007=17187067 

THE  proof  of  Multiplication,    is  bed   done   by 
Divifion. 


CHAP.     VI. 

Of  DIVISION  of  SIMPLE  NUMBERS. 

IVISION  is  a  fpeedy  method  of  fubtrafting 
one  number  from  another ;  to  know  how  many 
times  one  number  is  contained  in  another  ;  and  alfo 
\vhat  remains. 

THERE  are  three  requifites in  Divifion;  thedivifor; 
the  dividend,  and  the  quotient;  which  fhews  how 
many  times  the  divifor  is  contained  in  the  dividend. 

WHEN  any  number  meafures  another,  the  number 
fo  meafured,  is  faid  to  be  a  multiple  of  the  other  : 
thus,  21  is  meafured  by  7,  for  7  is  contained  juil  3 
times  in  21  ;  confequently  21  is  a  multiple  of  7. 

ONE  number  is  faid  to  meafure  another,  by  a  third 
number,  when  it- either  multiplies,  or  is  multiplied 
by  the  meafnring  number,  produces  the  number 
meafured.  (See  Euc'/id's  7th  book,  def.  23.) 

HENCE  it  follows,  that  in  Divifion  the  quotient 
rnuft  be  fuch  a  number,  which  if  multiplied  with  the 
divifor,  will  produce  the  dividend  ;  confequently 

Divifion 


(      37       ) 

Divifion  is  the  reverfe  of  Multiplication  ;  and  there 
fore  operations  in  Divifion,  muft  be  performed  direct 
ly  reverfe  of  thofe  in  Multiplication  -,  that  is,  the  di- 
vifor  muft  be  placed  firft ;  then  make  a  flroke  on  the 
right-hancfcof  it,  and  fet  down  your  dividend,  on  the 
right-hand  of  which,  make  another  ftroke,  to  feper- 
ate  the  dividend  from  the  quotient  ;  then  begin  on 
the  left-hand,  and  decreaie  the  dividend  by  a  re 
peated  fubtra&ion  of  the  products  of  the  divifor  and 
each  quotient  figure,  as  they  become  known. 

EXAMPLES, 

REQUIRED  to  divide  344  by  4;  the  operation  of 
which  will  Hand  in  the  following  order, 

dividend 
divifor  4^344(86  quotient 


24 

24 

oo 


THE  explanation  of  the  above  is  as  follows  :  firft 
enquire  how  many  times  your  divifor,  which  confifts 
of  i  figure,  is  contained  in  the  firft  figure  of  your  divi 
dend,  which  is  o  times  ;  becaufe  your  divifor  (4)  is 
greater,  than  the  firfl  figure  of  your  dividend  (3),  as 
appears  by  inflection  ;  and  therefore  cannot  meafure 
it ;  for  a  greater  number  to  meafure  a  lefs  is  abfurd ; 
therefore  you  muft  increafe  the  value  of  the  firft 
figure  of  the  dividend,  by  taking  the  annexed  figure 
(4)  into  the  expreflion  ;  which  will  then  be  34  (for 
the  reafons  before  given)  ;  then  enquire  how  many 
times  your  divifor  is  contained  in  thofe  two  figures 

of 


(       38       ) 

of  the  dividend,  to  wit,  34 ;  which  is  8  times,  for  8 
times  4  is  32,  and  32  being  the  greatefl  multiple  of 
the  divifor  that  can  be  made  under  34  -,  confequent- 
ly  8  mult  be  the  firft  figure  of  the  quotient,  which 
place  as  in  the  example  -y  then  multiplying  the  quo 
tient  figure  (8)  with  your  divifor,  as  in  Multiplica 
tion,  fubtract  their  product  from  thofe  two  figures  of 
the  dividend,  by  which  the  faid  quotient  figure  was 
obtained  ;  and  to  the  remainder  (2)  annex  the  next 
figure  of  your  dividend  (4),  and  the  remainder  fo  in- 
creafed  becomes  24 ;  then  enquire  how  many  times 
4  is  contained  in  24,  which  is  6  times  •>  therefore 
place  6  in  the  quotient,  and  multiply  it  with  your 
divifor,  fubtracting  their  product  as  before,  and  the 
work  is  done.  (See  the  example  page  37.) 

Now  the  quotient  obtained  in  the  example  is  86  -} 
and  there  being  no  remainder,  fhews  that  4  is  con 
tained  in  344,  juft  86  times. 

THE  greateft  difficulty  in  divifion,  is  when  your 
divifor  confifls  of  many  places  of  figures,  and  does 
not  exactly  meafure  the  figures  of  the  dividend  with 
which  you  compare  it :  therefore  to  find  the  right 
quotient  figure,  may  be  done  by  confidering  that 
the  product  of  the  quotient  figure  with  your  divifor, 
rouft  never  be  greater  than  that  part  of  the  dividend, 
with  which  you  compare  it ;  nor  yet  fo  fmall,  that 
the  number  remaining  after  fubtracting  the  product 
of  the  quotient  figure  and  divifor  from  the  aforefaid 
part  of  the  dividend,  fhall  be  greater  than  the  divifor. 
Therefore  by  fuppofing  a  figure  for  the  quotient,  and 
multiplying  it  with  a  figure  or  two  on  the  left-hand 
of  your  divifor,  you  may  eafily  determine  the  right  quo 
tient  figure;  which  may  be  obtained  by  fuch  mental 
operations,  on  the  fecond  or  third  trial,  at  fartheft. 

BY  thoroughly  obferving  the  foregoing  directions, 
you  may  proceed  to  the  performance  of  the  following 

examples  5 


(      39      ) 

examples ;  wherein  we  fhall  prove  thofe  operations, 
performed  in  the  laft  chapter ;  in  order  to  which,  we 
{hall  begin  with  the  fecond  example;  taking  the  pro- 
duel:  of  the  factors  for  a  dividend,  and  the  multiplier 
for  a  divifor ;  and  proceed  as  before.  (See  the  oper 
ation  annexed.) 

dividend 


divifor  421  \  10268 19(2439  quotient 
)   84.2- •• 


3789 
3789    - 

,.oo 

Note,  //  will  be~l>:ji  to  point  the  figures  of  the  divi 
dend,  as  they  are  annexed  to  the  fever al  remain 
ders  •>  without  which  you  may  annex  a  wrong  one. 

HERE  you  may  fee  the  quotient  is  the  fame  as  the 
multiplicand  of  the  example  before  quoted  ;  which 
proves  that  the  product  of  2439X42.1:1:1026819. 

Required  to  divide  779352354  by  32001. 


OPER- 


(    jt-o ) 

OPERATION. 

32001^779352354(24354:1:779352354—32001 
/ 64002 

139332 
128004 

113283 
96003 


172805 
160005 

12800.4 
128004 

.....  o 

Again,  divide  1798836  by  26. 
OPERATION'. 

26^1798836(69186—  1798836—26 
Ji  56 

238 


48 


223 
208 

~6 
i$6 

-   •*-  Once 


(       41       ) 

Once  more,  divide  12660228  by  98, 

OPERATION. 
98 \ 1 2660228(1 291  %6— quotient  required. 

286 
196 

900 
882 

182 
9* 

842 
784 

588 
588 

IF  there  be  cyphers  annexed  to  the  divifor  and 
dividend,  expunge  an  equal  number  in  both  faftors  : 
as  in  the  following  example. 

Divide  694000  by  2000. 

OPERATION. 
2(000^694(000(347^:694000-1-2000 


14 
14 


F  IT 


(       42       ) 

IT  will  fometimes  happen  in  Divifion,  that  the 
remainder,  when  augmented  by  annexing  the  next 
figure  of  the  dividend,  is  lefs  than  the  divifor,  and 
confequently  cannot  be  meafured  by  it;  in  which 
cafe,  place  o  in  the  quotient,  and  annex  the  next 
figure  of  the  dividend  to  the  former  number  ;  but 
if  this  number  be  ftill  lefs  than  the  divifor,  place  o 
in  the  quotient  and  annex  another  figure  of  the  div 
idend  ;  and  fo  on,  in  like  manner  till  the  faid  num 
ber  be  fo  in-creafed,  that  it  may  be  meafured  by  the 
divifor.  (See  this  illuftrated  in  the  following.) 

Divide  98953302  by  1302. 

OPERATION. 

1302^98953302(760011198953302-7-1302 
' 


...o 


THE  proof  of  the  remaining  examples  in  Multi 
plication,  are  left  to  the  fagacity  of  the  learner. 

It  is  required  to  divide  32176432  by  3476. 


OPER- 


(       43       ) 

OPERATION. 

3476^32176432(9256 
^31284 

8924 
6952 

19713 
17380 

23432 
20856 

2576  remainder. 

HERE  follows  fome  examples  and  their  anfwers 
without  their  work. 

What  is  the  quotient  of  23884044718-7-45007  ? 
Anfwer.     530674. 

What  is  the  quotient  of  34500000-7-100000  ? 
Anfwer.     345. 

What  is  the  quotient  of  24457  2OOO-T-J56  ? 
Anfwer.     687000. 

What  is  the  quotient  of  1332250-7-365  ? 
Anfwer,     3650. 

THAT  DivifiooJs  a  fpeedy  method  of  fubtra&ion, 
as  before  hinted,  may ^be  thus  proved.  Suppofe  18 
were  to  be  divided  by  6  :  firft  fubtraft  the  divifor 
from  the  dividend,  and  the  divifor  again  from  that 
remainder,  and  fo  on  till  nothing  remains.  (See  the 
operation  in  the  next  page.) 

OPER- 


(       44       ) 

OPERATION. 

1 8  dividend 

—  6  divifor 

12  remainder 

—6  divifor 

6  remainder 

—  6  divifor 


HENCE  it  is  manifeft,  that  the  divifor  is  contained 
in  the  dividend,  jufl  3  times  -,  that  is,  3  times  6:1:18  : 
confequently,  &c.  ^.  E.  D. 

THE  next  thing  to  be  confidered,  is  the  proof  of 
your  work,  i.  e.  whether  the  quotient  found  is  a  true 
one.  The  method  is  directly  reverie  of  that  ufed 
for  the  proof  of  Multiplication ;  for,  as  the  truth  of 
Multiplication  is  known  by  Divifion,  fo  that  of  Di- 
vilion  is  known  by  Multiplication  -,  that  is,  by  mul 
tiplying  the  quotient  with  the  divifor,  which  product 
inuft  be  equal  to  the  dividend  -3  therefore  multiply 
the  quotient  with  the  divifor,  and  to  their  product 
add  what  remains  after  divifion;  which  aggregate 
will  be  equal  to  the  dividend,  if  the  work  is  right. 

THERE  is  another  method  of  proving  Divifion  ; 
which  is  much  fhorter  than  the  former,  and  is  no 
more  than  adding  together  the  products  of  the  fever- 
al  quotient  figures  with  the  divifor,  as  they  (land 
in  your  operation  ;  which  aggregate,  together  with 
the  remainder,  will  be  equal  to  the  dividend.  (See 
the  following  example.) 

Required  to  divide  8765452  by  3463. 

OPER- 


(       45       ) 

OPERATION. 

3463)8765452(2531 

+  6926-    '   —3463X2000 

18394 
+  I73IS       1=3463X500 

10795 
+  10389     =3463X30 

4062 
+  3463  =3463X1 

+  599  remainder 


8765452= dividend. 


0,6926000+1731500+103890+3463  +  599:= 
8765452.     Therefore,  &c. 


A  SUPPLEMENT  to  CHAPTER  VI. 

NOTWITHSTANDING  what  hath  been 
faid  on  this  fubject,  refpecting  the  divifion  of 
fimple  quantities,  is  univerfally  true ;  yet  there  is 
another  method  of  dividing  quantities,  which  is  very 
ready  in  practice  ;  and  is  therefore  called  Short  Divi 
fion  :  this  method  is  performed  by  the  following  Rules. 

RULE      I. 

ARRANGE  the  factors  as  before  in  Divifion  ;  then 
by  comparing  the  divifor  with  the  dividend,  you 

will 


(      46       ) 

will  obtain  a  quotient  figure.,  which  muft  be  let  in 
its  proper  place,  under  that  part  of  the  dividend  by 
which  your  divifor  was  compared  ;  valuing  faid  fig 
ure  as  though  there  were  no  other ;  alfo  obtain  the 
difference  (if  any)  of  the  product  of  the  divifor  and 
quotient  figure,  and  the  aforefaid  part  of  the  divi 
dend  j  prefixing  that  difference  in  your  mind  to  the 
next  figure  of  your  dividend  ;  which  forms  an  expref- 
fion  for  obtaining  the  next  quotient  figure,  which 
muft  be  fet  directly  under  that  figure,  to  which  the 
difference  was  prefixed ;  and  fo  on  till  the  whole 
be  completed. 

EXAMPLES. 
Divide  46782  by  3. 

THOSE   numbers  being   placed   as  directed  will 
ftand  thus, 

3)46782 

15594=46782-3 

Again,  divide  68432  by  4  : 
thus,  4^)68432 

17 iQ%~quotient  required. 

Note  i.  If  there  be  a  remainder  after  the  lafl  quo 
tient  figure  is  found,  Jet  it  at  a  little  diflance  on  the 
right -hand  of  your  quotient,  making  a  dot  with  your 
•pen,  denoting  the  federation  -,  as  in  the  following. 


Divide  13764  by  5  :  thus,  5  \  23764 

/          '\ 


(    J-7 ) 

Again,  find  the  quotient  of  732 r  5—6: 
thus,  6x73215 


1 2  20  2  . 


Alfo,  divide  43206  by  8  : 

thus,  8  \432io6 

54013  .  zrem, 

Note  2.  If  your  divifor  be  10,  federate  the  fir  ft  fig 
ure  on  the  right-hand  of  your  dividend  for  a  re 
mainder,  and  the  work  is  done. 

thus,  10^76435(2^^. 

Find  the  quotient  of  645384-^-12: 
thus,  12^645384 


RULE      II. 

1.  RESOLVE  your  divifor  into  feveral  parts  fuch, 
that  their  continued  product  fhall  be  equal  to  the 
given  divifor. 

2.  SUBSTITUTE  thofe  parts  fucceflively  as  divifors, 
in  the  following  manner,  viz.  divide  the  given  divi 
dend  by  one  of  thofe  parts,  now  called  divifors,  and 
the  refulting  quotient  by  another  of  thofe  divifors, 
and  fo  on  5  the  laft  quotient  arifing  by  fuch  divifors, 
will  be  the  quotient  required. 

EXAMPLE. 

Divide  2904  by  24, 

YOUR 


YOUR  ditflfor  refolved  into  parts  as  above  directed 
will  be,  either  8  and  3,  6  and  4,  or  1 2  and  2  -,  for 
8X3=24,  6X4—24)  or  12X2=245  therefore  let  the 
parts  be  6  and  4 ;  then  2904—6^484,  and  484-7-4 
~ 1 2i~quotient  required ;  and  if  the  others  be  tryed 
they  will  equally  fucceed. 


CHAP.     VIL 

ADDITION  of  COMPOUND  QUANTITIES  or 
NUMBERS. 

ADDITION  of  compound  quantities,  is  the  add 
ing  together  numbers  of  different  denomina 
tions,  fo  that  their  aggregate,  or  total  amount  may 
be  known.     The  operations  are  performed  by  the 
following  general 

R  UL  E. 

1.  WRITE  down  the  feveral   denominations  fo, 
that  all  thofe  of  the  fame  name  may  ftand  directly 
under  each  other. 

2.  BEGIN  on  the  right-hand,  at  the  leaft  of  the 
given  denominations,  adding  together  the  whole  of 
that  denomination,  as  in  Simple  Addition ;  then  di 
vide  this  fum  by  fuch  a  number,  as  it  takes  parts  to 
make  one  of  the  next  greater  denomination,  placing 
the  remainder  (if  any)  under  its  own  denomination, 
and  carrying  the  quotient  to  the  faid  next  greater 
denomination,  add  them  up  with  the  whole  of  that 
denomination,  theri  divide,  as  before;  and  fo  on, 
from  one  denomination  to  another,  until  the  whole 
be  completed. 

SECT. 


(       49       ) 

SECT.       I. 

ADDITION  of  TROT  WEIGHT. 

TROY  WEIGHT  is  that  by  which  gold,  filver, 
jewels,  medical  compofitions,  and  all  liquors  are 
weighed.  It  is  divided  into  four  denominations, 
to  wit,  ft.  pounds,  oz.  ounces,  dwt.  pennyweights 
and  gr.  grains,  according  to  the  following 

TABLE. 


.24Odfw/.:ni  2  cz.ni  ft . 

EXAMPLES. 

Find  the  fum  of  the  following,  i/j-tb.  i  loz.  i6dwt. 
I3^r-  +  i9l6.  iooz.  IT  dwt.  ijgr.  +  ijfi).  noz. 
iidwt.  22gr. 

THESE  numbers  being  placed,  according  as  the 
general  rule  directs,  will  Hand 

ft.    oz.    dwt.  gr. 
fi4     ii      16     13 
thusX  19     10     17      17 

1          II       12       2± 


52     10       7       4^r jto  required. 

_  i 

THEN  begin  at  the  lead  denomination,  to  wit,  grains, 
and  adding  together  all  that  denomination,  we  find 
the  fum  to  be  52  :  now  becaufe  24  grains  make  one 
pennyweight,  divide  52  by  24,  and  the  quotient  will 
be  2,  leaving  a  remainder  of  4,  which  write  under 
grains,  and  carry  the  quotient  2,  to  the  next  place, 
and  adding  it  up  with  that  denomination,  we  find  the 
fum  to  be  47,  which  divide  by  20  (becaufe  20 
pennyweights  make  one  ounce)  and  the  quotient  will 

G  be 


be  2j  leaving  a  remainder  7,  which  write  in  its  prop 
er  place,  and  carry  the  quotient  2.,  to  the  next  place  ; 
this  being  added  up  with  the  denomination,  we  find 
the  Ium  to  be  34,  which  divided  by  12  quotes  2,  and 
10  remaining;  write  this  under  its  own  denomina 
tion,  and  cany  the  quotient  2  to  the  next  place,  which 
added  up  with  that  denomination,  we  find  the  furn 
to  be  525  and  becaufe  this  is  the  lad  denomination, 
write  the  whole,  and  the  work  is  done.  Hence  we 
find  the  fum  total  to  be  5216.  10  oz.  *]dwt.  and  4gr. 
as  was  required.  (See  the  example,  page  49,) 
Jb.  oz:.  dwt*  gr.  IJj.  oz.  dwt.  gr. 

37     I0     J7     19  47     IX     *9     24 

12      7     12     17  27       8     17     20 

17  10       17       12  19         7       12       17 

18  9     19     23  10      5     15   -17 


SEC  T,      JL 

ADDITION  of  MONET. 

*THIS  is  to  find  the  aggregate,  or  ium  total  of  fcv- 
tral  (urns  of  money. 

EVERY  nation  of  the  world  has  a  particular  method 
of  reckoning  their  money.  Great-Britain  makes  ufc 
of  pounds,  fhillings,  pence  and  farthings  3  and  the 
United  States  followed  the  fame  method,  until  the 
prefent  fyflem  of  government  was  eflablillied ;  by 
which  it  is  enadlcd,  that  all  the  monies  of  every  na 
tion  or  kingdom,  fhall  be  reckoned  or  eftimated  in 
America,  in  dollars  and  cents:  fo  that  thefe  two 
fpecies  of  money  are  to  be  made  the  ftandard  money 
of  the  United  States. 

Hitis  thai   100  c$nt$  m.akt  om  dollar. 

-RX/W 


(      5*       ) 


EXAMPLES. 

Find  the  Him  of  ij^dol.  I'-jcts.  -\-\y-doL  \gcts.-\~ 
37$^0/.  yicts.  -\-2j$dol.  yicts.  Thefe  being  plage4 
according  to  the  general  rule,  will  ftand 

cts. 


1023     20  j "urn  required. 

Note.  Since  100  ^tf/j  make  one  dollar,  we  mufl  di 
vide  thejum  of  the  cents  ly  100  ;  but  to  divide  by 
100  is  no  more  than  tojeperate  the  two  right-hand 
figures  of  the  dividend  for  a  remainder,  the  refl 
are  the  quotient,  ^herefore^  after  you  have  add 
ed  up  the  laft  flace  of  figures  in  the  cents'  place, 
proceed  to  the  dollars'  place  as  though  the  whole 
was  but  one  denomination. 


Find  the  fum  of  i  zjdol.  v        .     , 
$37 do!,  igcts.+  izzdol.  yicts. -\-ii7 dot.  yocts* 

do  I.  cts. 

127  19 

278  19 

137  19 

122  92 

1 27  90. 

793  39=/^m  required. 


dol.  cts.  dol.  cts.  doL  cts. 

127  17  3787  19  2784  19 

172  57  3729  72  1234  27 

189  68  4229  91  3456  78 

Total  ZZZZH       HHZZI       ZZZZZ 

HAVING  thus  explained  the  principles,  and  given 
a  general  rule  for  the  Addition  of  all  compounds  in 
whole  numbers  ;  we  lhall  leave  the  reft  to  the  faga- 
city  of  the  learner,  who  with  the  affiftahce  of  the  fol 
lowing  tables  and  examples,  will  be  able  to  manage 
any  fuch  compounds  as  have  relation  therewith. 

SECT.     III. 

Of  AVOIRDUPOIS  WEIGHT. 

By  Avoirdupois  Weight  are  weighed,  flefh,  but 
ter,  cheefe,  fait  ;  alfo  all  coarfe  and  drofTy  commodi 
ties  i  as  grocery  wares;  likewife  pitch,  tar,  rofin, 
wax,  iron,  flee],  copper,  brafs,  tin,  lead,  hemp,  flax, 
tobacco,  &c. 

THE  characters  in  Avoirdupois  Weight  zxtdr.  oz. 
Ib.  qr.  C.  cf.  that  is  drachm,  ounce,  pound,  quarter, 
hundred,  tun. 

TABLE. 

16  dr.-=.\  oz.     256^.—  i60%.=i/£.     7168^.= 
i£r.     2867  2  dr.-=^\  79202;.=! 

57344O  ^.=35840  6>2;.^z 


EXAMPLES. 

?.    C.  qr.  Ib.  oz.  dr.          T.    C.  qr.  Ib.  oz.  dr. 

346  12  2  16  10  14    576  19  i  16  12  13 

67  16  3  22  8  10    867  4  o  24  14  13 

4"6  10  3  15  12  15    453  6  3  27  3  4 


SECT. 


(       53      ) 

SEC  T.       IV. 

Of  APOTHECARIES  HEIGHT. 

THE  Apothecaries  pound  and  ounce  is  the  fame 
as  the  pound  and  ounce  Troy,  but  differently  di 
vided,  as  in  the  following 

TABLE. 

20  jr.— 13.     6o£T.=39.=i3.     48037.  =  249. 


APOTHECARIES  make  ufe  of  thefe  weights  in  the 
compofition  or  mixture  of  their  medicines,  but  fell 
their  drugs  by  Avoirdupois  Weight. 

EXAMPLES. 

K-     I-    5-  3-  Zr-          K>-     I-    3-  9-  Zr- 
124     10     4     2     14         266       9     5     i     15 

64       8     6     i     16  76     10     4     2     14 

30     ii     7     o     17  96     ii     6     2     10 

50       9     3     i     12  10     7     i       i 


SECT.      V. 

BY  Long  Meafure,  is  eftimaied  length,  where  no 
regard  is  had  to  breadth  :  or  in  other  words,  it  meaf- 
ures  the  diilance  of  one  thing  from  another  :  and  the 
ufual  method  of  dividing  and  fub-dividmg  of  length, 
is  into  degrees,  leagues,  mile's,  furlongs,  poles,  yards, 
feet,  inches,  and  barley-corns,  as  in  the  following 

TABLE     L 


23760^. 
190080^. 


(       54       ) 


^63-360  fa.^$2%of.=i76oyd.^32op.==&fur.==:  i  m. 

570240^.=! 


TABLE    II 


zifur.  1  90080^. 
=63360  /#.=528o/.=i  7  60^.=  i  60  t^.=i3/ar,=:  i  ^. 
570240  ^.=1  90080  f».=i  5  840/.==528ojK^.==48or£, 
=24/«r.=2  w.=  i  /^.  1  1  404800  ^f.  =38oi6oo///.= 
3  1  68oo/.=  105600^=^9600  -cb.=.4.%ofur.=  60  ;».== 


THE  length  of  a  degree  as  laid  down  in  table  ad. 
is  not  to  be  underftood  as  the  true  one,  but  the  length 
of  a  degree  as  commonly  received  and  pra6lifed  ;  for 
the  length  of  the  greatefl  degree  is  yo^Vrniles,  and 
the  leaft  67^.  miles  nearly;  a  mean  degree  is  there-' 
fore  6  8-^  miles. 

EXAMPLES. 
deg.     h.    m.fur.  cb.  yd.  f.    in,    be* 

I2O       14       2       6       14       5       2       1O       I 

87     12    o    7     12     3     i       50 
90     19     i     5     18     2     2       42 


332     15     i     7     12  8     i     10     2 

19     a    o     14  9     2       9 

i     6     13  5     o 

4     10  4 

9 


S  E  C  TV 


(      55       ) 

SECT.      VI. 

Of  LAND  MEASURE. 

THE  ufe  of  this  meafure,  is  to  find  the  area  or  fu- 
perficial  content  of  any  piece  of  land  in  acres,  and 
pares  of  an  acre  -3  which  parts  are  as  in  the  following 

TABLE. 


I  .     .fy.yd.—iofq.  cb.=ifq.  qr.     43560 

f-  —~  ,c/£.  fl&.=4/j.  qr.^ifq.  acre* 


EXAMPLES. 

:b.  yd.     /.              ac.  qr.  cb.  yd.  f. 

'6     104  -  8             9.2     i  7  loo  7 

j     7     1  1  1     7             27     3  7  98  8 

24       90     7              39     o  7  117  7 


SECT.       VII, 

Of  CLOTH  MEASURE. 

THE  divifions  of  Cloth  Meafure  are  as  in  the 
foyowing 

TABLE. 

Ailb,  j^r.s=i^// 


EXAMPLES. 

yd.      qr.  .  «rt.  <•//  F/.    qr.    na. 

226       3       2  3-733 

74       3       o  39       2       i  ' 

362       2       3  500       3       2 


$11  En%. 

qr. 

na. 

ell  Fr. 

jr. 

»*, 

3^7 

4 

3 

529 

5 

3 

90 

3 

2 

468 

2 

2 

264 

2 

I 

436 

4 

3 

354 

3 

3 

43 

3 

i 

SECT.      VIII. 

Of  DRT  MEASURE. 

DRY  MEASURE  is  fo  called  becaufe  it  meafurcs  all 
fuch  dry  commodities  as  corn,  .wheat,  rye,  oats,  bar- 
ley,  peas,  beans,  and  all  kinds  of  grafs-feed  -,  alfo  all 
kinds  of  roots  and  fruits. 

TnEftandard  of  thismeafnre,  is  abufhcl  of  a  cylin 
drical  form,  of  the  following  dimenfions,  viz.  1 8~  in 
ches  in  diameter,  and  8  inches  in  altitude  $  confe- 
quently  a  veflel  of  fuch  form  and  dimenfions  will 
contain  2150^^  cubic  inches,  which  is  the  content 
of  the  Wincheiter  bufhel:  Therefore  the  quart  Dry 
Meafure,  contains  67-^5-  cubic  inches  nearly  ;  and  the 
divifions  are  as  in  the  following 

TABLE. 

67  .  2  cu.  tn.^iqrt.  268  .  8  cu.  in. =4  qrt.^i  gal. 
537  .6  cu,  in.i=&  qrt.~zgaL-=.ipc.  2150.42^.  in. 
zzz^iqrt.—Kgal.-^^fc^i  bujh. 

EXAMPLES, 
lujh.  pc.  gal.  qrt.    bujh.  pc.  gal.  qrt.  bu/h.  pc.  gaLqrt. 

57     3     i     3         37     3     *     l         2312 
24     002         19000  it 

47     210         33     203         2313 

SECT. 


(      57      ) 


SECT.      IX. 

Of  LIQUID  MEASURES. 

IN  Liquid  Meafures,  the  gallon  is  made  the  (land- 
ard,  and  from  thence  are  deduced  the  other  denomi 
nations  made  ufe  of  in  fuch  meafures.  The  wine  gal 
lon  is  fuppofed  to  contain  231  cubic  inches,  confe- 
quently  the  quart  muft  contain  57^  cubic  inches  j 
from  thence  is  deduced  the  following 

TABLE  of  WINE  MEASURE. 

$7^eu.in.~iqrt.  231  r#.  in.-zz^qrt.^i  gal.  9702 
cu.  /#.=i68  qrt.=z4.2gal.-=itr.  14553  cu.in.-=.2$2qrt. 
=63£#/~i  ^  tr.=i  bhd.  19404  cu.  /».—  336^77  .=84 
g0/.=:2  tr.~i^  bbd.—i  pun.  29106  cu.  ^.=504  qrt. 
==126  gal.^z  /r.=2  bbd.~i*i-fun.'=.i  bt.  58212  cu. 


i  tun. 

EXAMPLES. 

tun     bbd.  gal.  qrt.  tun.  bbd.  gal.  qrt. 

237       2  62  3  279  2  57  2 

234       i  27  o  273  o  39  o 

72       2  25  3  99  2  47  3 

34      o  59  o  93  i  24  2 


Of  ALE  or  BEER  MEASURE. 

THE  gallon  of  Ale  or  Beer  Meafure  contains  282 
cubic  inches,  as  in  the  following 

TABLE. 

*1&^cu.in.—  iqrt.  282  *«.  f#.=4.  qrt.— i  gal.  2397 
cu.  *».=34  qrt.-=$)Lgal.-=>\Jir.  4794  £#.  /'#.=-68  ^r/. 
=17  £tf/.=2/r.=i  A//.  9588  c^.  ;/;.=:i36  ^.=34 

H  gal. 


.=i  ^r.     14382  r^. 
kil.—i±  bar.~i  bbd. 


EXAMPLES. 

bbd.  kit.  fir.    gal.  qrt.  bhd.  kit.  fir.    gal.  qrt. 

79  2  i       7       2  73  2       i       6       3 

64  3  o      5  ;j  o  97  i       i       7       2 

49  i  162  37  2120 


SECT.      X. 

O/  the  MEASURE  of  flME. 

IN  the  divifion  of  Time,  a  year  is  made  thefland- 
ard  or  integer,  which  is  determined  by  the  revolu 
tion  of  fome  celeftial  vbody  in  its  orbit ;  which  body 
is  either  the  fun  or  moon.  The  time  meafured  by 
the  fun's  revolution  in  the  "ecliptic  (or  imaginary 
circle  in  the  heavens,  fo  called  by  aftronomers)  from 
any  equinox  or  foltice  to  the  fame  again,  is  365  days, 
5  hours,  48  minutes,  57  feconds,  and  is  called  the 

iblar  or  tropical  year. Although  the  folar  year 

before  mentioned,  is  the  only  proper  or  natural  year, 
yet  the  civil  or  Julian  year  is  the  one  which  the  dif 
ferent  nations  of  the  world  make  ufe  of  in  the  re 
gulation  of  civil  affairs. 

THE  civil  folar  year  contains  365  days,  6  hours  ; 
but  in  common  mathematical  computations,  the  odd 
hours  are  generally  negledled,  and  the  year  taken 
only  for  365  days;  from  which,  the  divifions  in  the 
following  TABLE  are  made,  wherein  a  fecond  is  con- 
fidered  (as  it  really  is)  the  lead  part  of  time  that  can 
be  truly  meafured  by  any  mechanical  engine. 

6o*.-=i/.      36oo//.=6o/.=i 
?4&.=:i  d.     3i536ooo//.=^ 
^i  year.  EXAM- 


(59   ) 

EXAMPLES. 

y.  d.   h.   l  "  y.   d.  h.   '  " 

167  272  14  42  29  173  192  10  17  29 

234  i?3  22  58  59  346  364  23  59  59 

39  290  19  19  19  199  170  19  17  16 

43  222  22  22  22  344   19  IO  34  46 

99  99  20  57  21      79  38  23  43  43 


SECT.      XL 

Of  CIRCULAR  MOTION. 

WHAT  is  here  meant  by  Circular  Motion,  is  that  of 
the  heavenly  bodies  in  their  orbits  ;  which  are  reck 
oned  in  figns,  degrees,  minutes,  and  feconds,  as  in 
the  following 

TABLE. 


/=6o/=io.  io8ooo//.=i8oo/=30d 
=16".  1  296ooo"=2  1  6oo/=36o°=i  2  S*  —great  circle 
of  the  ecliptic  * 

EXAMPLES. 
S.      °        *  :      °  S.      °        '        ^ 

10      12      30      10  II       13       13       IJ 

9     ii     47     47  8     17     23     43 

4     37       4  7     29     44     27 

7     24    42     36  6     19     38     59 


Note.  In  the  Addition  of  Circular  Motion*  when- 
the  Jum  of  the  Jigns  exceed  1 2,  or  any  inultifle  of 
if,  writs  fuch  excefs  in  the  place  of  Jigns ,  rejecting 
tht  reft. 

Note. 


Note.  In  order  to  prevent  a  mijconftruftion  of  the 
abbreviations ,  in  the  nine  preceding  TABLES,  ive 
have  Jubjoined  the  following  explanation,  viz.  gr. 
Jiands  for  grains.      ^Jcruples.     %  drachms,      g 
ounces,    fl3  pounds. — be.  barley-corns,    in.  inches. 
f.  feet.    yd.  yards,     ch.  chains,     p.  poles,    fur. 
furlongs,    m.  miles,    le.  leagues,    deg.  degrees. — 
fq.Jquare.     qr.  quarters,     ac.  acres. — na.  nails. 
Flern.  Flemijh.    Eng.  Englijh.    Fr.  French. — cu. 
cubic. — qrt.  quarts,  gal. gallons,  pc. pecks,  bufh. 
bufhels. — tr.  tierces,    hhd.  hog/heads,  pun.  punch 
eons,    bt.  butts. —fir. firkins,   kil.  kilderkins,  bar. 
barrels. — "feconds.   *  minutes,    h.  hours,   d.days. 
y. years.     °  degrees.     S.  Signs. 


CHAP.      VIII. 

SUBTRACTION  of  COMPOUNDS.  ' 

SUBTRACTION  of  Compounds  is   the  taking 
o;ie  number  from  another:  and  is  performed  by 
the  following  general 

R  ULE. 

1.  RANGE  the  given  denominations  according  to 
the  dire&ions  in  the  laft  chapter. 

2.  BEGINT  at  the  fame  place  as  in  Addition,  to  wit, 
at  theleaftof  the  given  denominations,  fubtra&ing  the 
lower  number  from  the  upper,  as  in  Simple  Subtrac 
tion,  writing  the  difference  under  its  own  name ;  but 
if  the  number  in  the  fubtrahend  or  under  number,  be 
greater  than  that  which  (lands  direftly  over  it  (as  it 
prten  happens)  you  mult  add  to  your  upper  number, 
fo  many  units  of  that  denomination  as  are  equal  to 

one 


one  of  the  next  greater  ;  from  which  perform  thein- 
tended  fubtmetion,  writing  the  difference  as  before. 
Then  proceed  to  the  next  place,  where  you  muft  pay 
what  you  before  borrowed  of  this  denomination,  by 
adding  one  to  the  fubtrahend,  and  then  perform  fub- 
tradtion  as  before  •>  and  fo  on  to  the  laft  place,  where 
the  fubtraclion  is  performed  as  in  fimple  quantities. 

EXAMPLES. 

From  37  ft  1002.  17  dwt.  2O£r.  take  27  ft  noz. 
19  dwt.  ijgr. 

Thefe  numbers  being  placed  according  to  the  rule, 
will  ftand 

ft  cz.  dwt.  gr. 
37  io  17  20 
27  ii  19  17 


9     io     1 8       3  diff.  required. 


HERE  beginning  at  the  leaft  denomination,  to  wit, 
at- grains,  fubtract  17  from  20,  and  there  remains 
3,  which  write  under  its  own  namej  then  pro 
ceed  to  the  next  denomination  -9  but  here  the  under 
number  is  the  greateft,  and  therefore  cannot  be  taken 
from  the  upper ;  wherefore  add  ao  to  the  upper  num«* 
ber  (becaufe  20  pennyweights  make  one  ounce)  and 
the  fum  is  37,  from  which  take  19,  and  their  remains 
1 8  ;  or  take  19  from  20,  and  then  add  17,  and  the 
fum  will  be  1 8,  as  before.  Then  proceed  to  the  next 
place ;  and  here  again,  the  under  number  is  the  great- 
eft,  therefore  add  I  to  1 1  for  what  you  before  borrowed, 
and  the  fum  will  be  12,  which  taken  from  22,  leaves 
io,  which  write  in  its  proper  place,  and  proceed  to 
the  laft  denomination,  where  paying  what  you  before 
borrowed,  perform  the  fubtraclion  as  in  whole  num 
bers, 


bers,  and  the  remainder  will  be  9.  Hence  we  find  the 
whole  difference  to  be  9  pounds,  10  ounces,  18  pen 
nyweights,  and  3  grains. 

ft      $z.  dwt.  gr.          doL    cts. 
From  27     10     13     17  37     19 

Take  22       8     19     19  21     18 


Rem.    5       i     13     22  16       i  51     17 


As  the  foregoing  rule  is  general,  the  learner  by  du 
ly  obferving  the  application  of  it,  to  the  above  exam 
ples,  may  very  readily  perform  the  following  ones 
without  any  further  direction. 


T.      C.    qr.  It.      cz.  dr. 
From  324     19     3     17       2     15 
Take  233     17     2     20     13     14 

yd.    qr.  na 
±27     3     2 
204     i     3 

Rem. 

ellFlem.qr.na.  ell  Bpg.  qr.  na.    e 
From  5213         42      4      i 
Take  35      2      i         36      2      3 

HFr.  qr.    na, 
53      3      3 
49      5      o 

Rem. 

T.  bhd.  gal.  qrt.         bhd.  kil. 
From  37     3     36     2             33     * 
Take  23     i     37     3             27     i 

Jlr.  gal  qrt. 
*     7     3 
043 

Hem. 

y.     d.     b.     '      '            ft 
From  434  320  17  24  42           47 
Take  329  370  19  47  29           45 

I   2  3  gr. 

10  7  2  14 

8  5  i  17 

Rem.  ~  '  ; 

THE 


_ 

THE  method  of  proving  your  work,  is  the  fame  as 
that  of  Simple  Subtraction. 


CHAP.     IX. 

MULTIPLICATION  and  DIVISION  of  COM 
POUNDS. 

SECT.       I. 

Of  MULTIPLICATION. 

MULTIPLICATION  of  Compound  Numbers 
is  the  multiplying  any  fum  compofed  of  divers 
denominations,  with  a  fimple  multiplier,  according 
to  the  following 

RULE. 

BEGIN  the  operation  as  in  all  other  compounds, 
anultiplying  that  denomination  with  your  multiplier, 
as  in  Simple  Multiplication  ;  then  divide  this  pro- 
duel  by  as  many  units  as  make  one  of  the  next  great 
er  denomination,  writing  the  remainder  as  in  Addi 
tion  ;  then  note  the  quotient,  and  proceed  to  the 
next  place,  and  multiply  that  denomination  with  your 
multiplier,  to  which  add  the  aforefaid  quotient ;  then 
divide  this  produd  as  before,  and  fo  on,  till  you  have 
multiplied  your  multiplier  with  every  denomination 
in  your  multiplicand  j  and  the  refult  will  be  the  pro- 
duel:  required. 

EXAMPLES. 

Multiply  120  16   10 02.   i$dwt.  i J gr.  with  4, 

OPER- 


OPERATION. 

fe     02.  dwt.  gr. 
1 20     10     13     17  multiplicand 
4  multiplier 


483       6     14     20  froduft  required. 

HERE  we  begin  with  4X17=^68  ;  then  68-^-24= 
2,  and  20  remaining,  which  write  in  its  proper  place ; 
then  4X13=52,  to  which  add  2,  the  quotient  juft 
found,  and  thefum  will  be  54;  then  54-7-20—2,  and 
14  remaining,  which  write  in  its  proper  place  ;  then 
4X  10=40,  to  which  add  the  laft  quotient  2,  and  the 
fumis42;  now42-r-i2=rr3,  and  6  remaining,  which 
write  in  its  proper  place*  Laftly,  4X120=480,  to 
which  add  3,  the  laft  found  quotient,  and  the  fum  is 
483.  Hence  we  find  the  whole  product  to  be  483 
pounds,  6  ounces,  14  pennyweights,  and  20  grains. 

Multiply  \<r]doL  ijcfs^  with  6. 

OPERATION.  ** 

doL    cts. 

127     17 

6 


$: 

10 

o          ' 

13     42 

IO 

4 

763       2  froduft. 


yd.  qr.  na. 
423 
6 


5     24    48     40  frod.        28     o 


ellFlem. 


ellFL  qr.  na.   ellEng.  qr.  na,     ell  Ft.  qr,  tfa. 
17     2     i  10    4     2  13     5     3 

7  12  8 


124     o     3         130    40         in     4    Qprod, 


deg.    le*    tn.fur.  p.     f.     in.  be. 

12       10       2       5       10       10       I       2 

4 
50      3     i     5       2      7     6     zprcduff* 


Note.  You  may  refolve  your  multiplier  into  feveral 
parts,  as  in  Short  Divifeon,  and  if  thofe  parts  when 
multiplied  together,  do  not  exaftly  make  the  gi*uen 
multiplier  y  add  as  many  times  the  multiplicand  to 
the  produft,  as  the  product  of  the  f aid  parts  fall 
Jhort  of  the  given  multiplier  ;  as  in  thefe : 

Find  the  produdt  of  127  doL  ipc/j-.XiS- 
HERE  the  parts  of  the  multiplier  are  3  and  j. 
Therefore,  |  f/7    '£ 

3 

33 1     57 

— ; doi. 


1907     85  (becaufe  3XS=»5)= 


Required  the  pro  dud  of  iy]  doL  87  £/ 

I  Lee 


(      66      ) 

et  the  parts  be  3  and  7.     Therefore, 

dol.    cts. 
197     87 
3 


593     6 1 
7 
doL    cfs. 


4155     27=197    87X21. 
add  2  times  iyjjol.  Sjcts.  or  395     74 

4551 


What  is  the  product  of  228)  6  oz.  isdwt.  iigr. 

X32  ? 
Anfwcr.     72i]b  4oz .  16  dwt. 

What  is  the  product  of  i^yd.  3  qr.  ina.y(^? 
Anjw  er.     666  yd. 

SECT.       II. 

DIVISION  of  COMPOUNDS. 

DIVISION  being  directly  the  reverfe  of  Multipli 
cation,  needs  no  other  explanation  than  the  follow 
ing  examples  ;  only  obferve,  that  when  any  denomi 
nation  is  not  exactly  meafured  by  the  divifor,  the  re 
mainder  mufl  be  reduced  to  the  next  inferiour  deno 
mination,  and  added  to  its  then  perform  thedivifion. 

EXAMPLES. 

lb   oz.  dwt.gr.  dol.  cts.  dol.   cts. 

2V?75   ii    13   14       4)347   12       7)784  49 

187    ii    1 6    1.9  86  78  112     *]  quo. 

k. 


(       67       ) 

o      le.  m.fur.  cb.  yd.  f.  in. 
4U7     14     2     6     12     5     i     7 

.  -4-6—527  ^/.  5o<rAr.  and  527^7. 


Likewife,  loidol.  50  £//.-:-  5=20  </07.  30  cts.  (be 
caufe  6X5X5=150^=3165-7-150 


Miscellaneous  ^uejtions  for  the  Learn 
er  s  Pratt  ice. 

SI  R  Ifaac  Newton  was  born  in  the  year  1642,  and 
died  in  1726  :  What  was  his  age  whence  died  r 

There  are  two  numbers,  the  greater  96,  and  the 
lefs  45  :  What  is  their  funi  and  difference  ? 

To  find  a  number  fuch,  that  426  taken  from  it, 
will  leave  127  remainder. 

A  certain  number  of  merchants  in  trade,  gained 
19140  dollars,  which  being  equally  divided,  a  fliare 
was  found  to  be  4785  dollars  :  How  many  merchants 
were  there  in  that  trade  ? 

What  is  the  quotient  of  3276  divided  by  3,  and  by9  f 

What  number  is  the  divifor  of  1530320,  when  the 
quotient  is  470  ? 

What  is  the  coft  of  51  yards  of  broadcloth,  at 
locts.  per  yard  ? 


CHAP, 


CHAP.      X. 

REDUCTION.. 

BYRedu&ion,  numbers  compoled  of  different  der 
nominations  are  brought  into  one,  by  unfolding 
the  feveral  denomination's  by  the  parts  that  compofc 
them.  Or,  from  any  number  of  homologous  parts, 
to  difcover  the  number  of  certain  heterogeneous,  or 
unlike  denominations.  The  former  is  called  Reduc 
tion  by  Multiplication,  and  the  latter  Reduction  by 
pivifion.'  -  Reduction  by  Multiplication  has  the 
following  general 

RULE. 

% 

BEGIN  at  the  greateft  denomination  mentioned, 
multiplying  it  with  as  many  units  as  one  of  this  de 
nomination  contains  units  of  the  next  inferiour  de- 
denomination  ;  and  to  the  product  add  the  numbers 
in  the  lefs  denomination  ;  then  multiply  this  fum  as 
before,  add  as  above,  and  fo  on  (multiplying  with  as 
many  units  as  it  takes  thofe  of  «the  next  lefs  denomi 
nation  to  make  one  of  the  prefent),  until  you  have 
reduced  the  given  parts  to  the  denomination  required. 

EXAMPLES. 

Required  the  number  of  cents  equal  to  IGCO  dollar, 
OPERATION. 

JOOO 


of  cent  3  in  a  dollar. 
iooooo~tumfar  of  cents  required. 

Reduce 


Jlediice  1057  dol.  90  as.  into  ctfnts 
OPERATION, 


cts. 
90 


ICO 


^r.umber  *f  cents  required. 


BUT  to  reduce  the  monies  of  foreign  nations,  to 
thofe  of  the  United  States,  confult  the  following 

TABLE. 

dol.  cts. 

Pound  Sterling  of  Great-Britain^^.    44 
Livre  Tcitrnois  of  Fran'ce  1 8-J 

Guilder  of  the  United  Netherlands         39 
Mark  Banco  of  Hamburgh  33-^- 

Rix  Dollar  of  Denmark  I 

Rix  Dollar  of  Sweden  i 
Real  Plate  of  Spain  10 

Milree  of  Portugal  I     24 

Pound  Sterling  of  Ireland  4     10 

Tale  of  China  I     4? 

Pagoda  of  India  I     94.^ 
##/><?£  0/  Bengal  5  54. 

Mexican  Dollar  §  j 

Crown  of  France  I     i  j 

Crewn  of  England  I     1 1 

M6te.  The  go  Id  coins  of  France,  England)  Spain,  and 
Portugal^  are  valued  at  89  ^#/,f  per  pennyweight. 


In 


(      7Q      ) 

In    127  pounds  fterling  of  Great-Britain,   how 
many  cents  ? 

Here  multiply  the  pounds  with  444. 
127 
444 

508 
508 
508 

56388 -the  anfuser. 


In  274  livtes  tournois  of  France,  how  many  cents? 

Multiply  with  18,  and  add  half  the  multiplicand 
to  that  produ6h 

274 
18 

2192 

274 

4932 
*37 

5069  the  an  fiver. 
In  540  marks  banco  of  Hamburg :  how  many  cents  ?' 

Multiply 


Multiply  with  33,  and  add  on$  third  of  the  oiulti 
plicand  to  that  produft. 

540 

33 


In  424  rupees  of  Bengal :  how  many  cents  ? 

Multiply  with  55,  and  proceed  as  in   the   livres 
tournois  of  France. 

424 
55 

2I2O 
2 1  2O 

23320 
112 

23532  the  anfwer. 

Note.  In  reducing  the  following /peeies  of  money  U 
cent s ^  take  the  following  methods. 

For  the  Guilders  of  the  United  Netherlands  >  multiply 

with  39 

Real  Plate  of  Spain  10 

Milree  of  Portugal  t  1 24 

Pound  Sterling  of  Ireland  '410 


fale  of  China  -  148 

Pagoda  of  Indit  1  94 

Crown  of  Francs  in 

Crown  of  England  in 

In  127  Jbj  how  many  ounces,  pennyweights  and 
grains  ? 

127 
i  i-^Jiumber  of  ounces  in  i  found 

of  ounces  in  127  founds 
of  fenny  weights  in  i  ounce 

of  fennyw  eights  in  \  27  founds 
of  grains  in  i  fenny  weight 


121920 
60960 


of  grains  in  127  founds. 


16.  oz.  dwt,  gr. 
In  1  2     8     12    4  how  many  grains  ? 

12 


2O 

3052=152x204-12 
24 

I22I2 
6104 

73252=3052x24  +4~nuwfor  of  grains  req. 

In 


(        73     ) 

In  333  milrees  of  Portugal :  how  many  cents  ? 
Anfwer.     41292. 

In  555  tales  of  China  :  how  many  cents  ? 
Anfwer.     82140. 

REDUCflON  by  DIVISION. 

THIS  method  is  dire&ly  reverfe  of  the  former  \  for 
where  we  before  multiplied,  here  we  muft  divide 
with  the  fame  number  -9  and  therefore  admits  of  the 
following 

RULE. 

DIVIDE  the  numbers  in  each  denomination,  by  the 
number  of  units  that  make  one  of  the  next  fuperiour 
denomination  ;  and  the  quotients  refulting,  will  be  the 
numbers  in  the  feveral  denominations  required, 

EXAMPLES. 

In  57200  cents  :  how  many  dollars  ? 
1(00^572(00 

Therefore  572  dollars  is  the  anfwer. 

In  73252  grains  Avoirdupois  :  how  many  penny 
weights,  ounces,  and  pounds  ? 

24^73252     203052 


^ 


12  152 ..  I2rem. 
125 
1 20  12.8  rem. 


48  * 

K  Therefore 


(       74       ) 

Therefore  in  73252  grains,  there  are  3052  penny 
weights,  152  ounces,  or  12  pounds. 

Note,  tfhejeveral  remainders  are  of  the  Jams  n'ame 
of  their  dividends. 

In  41292  cents:  how  many  milrees  of  Portugal? 
41292-7-12411:333,  the  anjwer. 

In  82140  cents  :  how  many  tales  of  China  ? 
Anjwer.     555. 

In  5^388  cents:  how  many  pounds  fterling  of 
England  ? 

Anjwer.     127. 

Note.  In  reducing  cents  into  livres  tournois  of  France, 
you  muft  multiply  with  2,  and  divide  that  pvodufl 

by  37  •< The  mark  banco  of  Hamburg,  multiply 

with  3,  and  divide  that  fro  du  ft  by  ioo.>      -  The 
rupee  of  Bengal,  multiply  with  2,  and  divide  bym. 

In  752  nails  :  how  many  yards  ? 
Anjw  er.     47  yards , 

In  15840  barley  cornc :  how  many  miles  ? 
Anfwer.     3  miles. 

In  469  gallons  :  how  many  hogfheads  ? 
Anfwer.     ^hbd.  3% gal. 


Mifcellaneous 

THE  comet  of  1680,  at  its  greateft  diftance  from 
the  fun,  was  11184768000  miles:  now  fup- 
pole  a  body  had  been  projected  from  the  fun,  with  a 
degree  of  fwiftnefs  equal  to  that  of  a  cannon  ball, 

which 


(      75      ) 

which  is  at  the  rate  of  480  miles  per  hour  :  in  what 
time  would  this  body  reach  the  aforefaid  comet  ;  al 
lowing  the  year  to  confift  of  365  days  ? 
Anjwer.     2660  years. 

How  many  times  will  a  fhip  of  97  feet  6  inches 
long,  fail  her  length,  in.  the  diftance  of  1 2800  leagues 
and  10  yards; 

Anfwer.     2079408. 

A  MERCHANT  bought  4  tuns,  15  hundreds,  and  24 
pounds  of  fugar,  and  ordered  it  to  be  put  up  into 
parcels  of  24  pounds,  of  20,  of  16,  of  1 2,  of  8,  of  4, 
of  2,  and  of  each  a  like  number.  How  many  parcels 
will  be  made  of  the  fugar  ? 

Anfwer.     1 24. 

A  GENTLEMAN  had  15  dollars  to  pay  among  his 
labourers — to  every  boy  he  gave  10  cents — to  every 
woman  20  cents,  and  to  every  man  45  cents :  the 
number  of  men,  women  and  boys  was  the  fame,  I 
demand  the  number  of  each  fort  ? 

Anfwer.     20. 

THERE  are  five  tooth  wheels  placed  in  fuch  order, 
that  their  teeth  play  direclly  into  each  other :  the  firft 
wheel  contains  500  teeth — thefecond75o — the  third 
1500 — the  fourth  2000,  and  the  fifth  3000:  how 
m^ny  times  will  the  fifth  wheel  turn  in  100  turns 
of  the  firft  ? 

Anfwsr.     600, 

THE  velocity  of  light  being  at  the  rate  of  10000000 
miles  per  minute,  takes  up  6  years,  32  days,  5  hours, 
and  20  minutes  in  coming  from  the  neareft  fixed  ftar 
to  the  earth :  what  is  the  diftance  of  that  ftar  ? 

Anfwsr*     32000000000000. 

PART, 


PART       JL 

CONTAINING    tHE    DOCTRJNE     OF 
VULGAR    FRACTIONS, 


CHAP.       I. 

DEFINITIONS  and  ILLUSTRATIONS. 

A  FRACTION  is  a  broken  quantity,  or  the 
parts  of  an  unit,  which  are  exprefTed  like  quan 
tities  in  divifion  ;  to  wit,  by  writing  two  quantities, 
one  above  and  the  other  below  a  fmall  line  $ 


thus  \lmmerator  prl'X?,,! 

3  \  4  denominator  or  divifor  $4  4 
which  rs  three  times  the  quotient  of  unity  divided  by 
4:  therefore  in  all  Vulgar  Fractions,  unity  is  divid 
ed  into  fuch  parts,  as  are  exprefied  by  the  denomina 
tor  ;  that  is,  the  denominator  exprefles  what  kind  of 
parts  the  unit  is  divided  into,  and  the  numerator 
the  number  of  thofe  parts. 

HENCE  it  follows,  that  all  Vulgar  Fra&ions  what- 
foever,  reprefent  tfye  quotients  of  quantities,  which 
are  to  unity,  as  the  numerator  to  the  denominator  -, 
thus,  if  the  fraction  be  ^,  it  will  be  4  :  i  :  :  3  :  4  ,  and 
fo  on  for  others. 

ALL  Vulgar  Fractions  whatfoever,  fall  under  the 
five  following  forms,  viz.  proper,  improper,  fingle, 
compounded,  and  mixed, 

A 


(       77       ) 

A  PROPER  fraction,  is  when  the  numerator  is  lefs 
then  the  denominator  :  thus  -J-,  -£,  and  X7T>  are  proper 
factions. 

AN  improper  fraction,  is  when  the  numerator  is 
greater  than  the  denominator  :  thus  £,  -f  ,  and  -^,  are 
improper  fractions. 

A  SINGLE  fradtion,  is  a  (imple  exprefilon  for  the 
parts  of  an  unit  :  thus  4-,  -'->  and-*-,  are  fmgle  fractions. 

A  COMPOUND  fraction,  is  a  fraction  of  a  fraction  : 
thus,  ~  of  -^  and.y  of  ^  of  4?  are  compound  fractions. 

WHSN  whole  numbers  are  joined  or  connected  with 
fractions,  they  .are  fometimes  called  mixed  numbers  ; 
as  ioi,  and  15  -§-. 

A  MIXED  fraction,  is  when  either  or  both  the  nume 
rator  and  denominator,  is  a  mixed  number  : 
r  j  2!  17—  T 

thus,  |  —  i  and  -~7j,  are.  mixed  fractions. 


ANY  whole  number  may  be  expreffed  in  the  form 
of  a  Vulgar  Fraction,  by  writing  unity,  or  I  under  it: 

thus,  i2Oz=  -  and  Cj2=£-  &c. 
i  i 

THE  commpn  meafure  of  two  numbers,  j^any 
number  that  will  meafure  both  without  a  remainder  : 
thus,  3  is  the  common  meafure  of  9  and  12  ;  becaufe 
it  meafures  9  by  3,  and  1  2  by  4. 

THE  greateft  common  meafure  of  two  numbers,  is 
the  greateft  number  that  will  meafure  both  without 
a  remainder  :  thus,  7  is  the  greateft  common  meafure 
of  21  and  49  ;  becaufe  no  number  greater  than  7  can 
meafure  21  and  49,  without  a  remainder. 

ANY  number  that  can  be  meafured  by  feveral  other 
numbers,  the  number  meafured,  is  called  their  com 
mon  multiple  :  thus,  24  is  a  common  multiple  of  4 
and  6,  for  2X12—24,  4X61124,  and  6X4—24  :  the 
leaft  number  that  can  be  meafured  in  this  manner,  is 

called 


(       78       ) 

callecl  the  lead  common  multiple:  thus,  12  is  the 
kail  common  multiple  of  4  and  6  ;  becaufe  no  num 
ber  lefsthan  12,  can  be  divided  by  4  and  6,  with 
out  a  remainder. 

A  PRIME  number  is  that,  which  is  meafured  only 
by  unity:  as  5,  7,  n,  19,  &c. 

NUMBERS  prime  to  each  other  are  fuch,  as  no  num 
ber  except  unity  will  meafure  both  without  a  remain 
der  :  thus,  9  and  4  are  numbers  prime  to  each  other ; 
for  although  2  will  meafure  4  without  a  remainder, 
yet  it  cannot  divide  9  without  a  remainder:  3  may 
meafure  9,  but  it  cannot  meafure  4 :  therefore,  &c. 

A  COMPOSED  number  is  that,  which  fome  certain 
number  meafures :  thus,  6,  8  and  12,  are  compofed 
numbers;  for  jx21^^,  4X2—8,  and  2x6=12. 


CHAP.      II. 

REDUCTION  of  VULGAR  FRACTIONS. 

REDUCTION  of  Vulgar  Fractions,  is  the  chang 
ing  of  one  fraction  into  another  of  equivalent 
value ;  and  thereby  fitting  them  for  the  purpofe  of 
Addition,  Subtradlion,  &;c. 

THE  whole  bufinefs  of  Reduction,  is  comprifed  in 
the  following  Problems. 

PROBLEM      I. 

To  find  the  lea/}  common  multiple  of  fever al  numbers. 

RULE. 

1.  RANGE  the  numbers  in  a  direct  line. 

2.  FIND   what  number  will  divide  two  or  more  of 
them  without  a  remainder;  by  \^iich  divide  them, 

and 


(      79      ) 

and  fet  their  quotients  together  with  the  undivided 
numbers,  in  a  line  beneath. 

3.  DIVIDE  this  line  in  the  fame  manner  as  the  firfl  $ 
and  fo  on,  from  line  to  line,  until  no  number,  except 
unity  will  divide  two  of  them  without  a  remainder  ; 
then  the  continued  product  of  all  the  divifors,  and 
the  lad  quotients,  will  be  the  Jeafl  common  multi 
ple  required. 

EXAMPLES. 
Find  the  leail  common  multiple  of  4,  8,  and  12, 

OPERATION. 
4U     8     12 


WHENCE,  4X*X2Xj  —  24,  the  leafl  common 
multiple  required,  ij 

Find  the  leafl  common  multiple  of  2,  4,  6,  7  and  20. 
OPERATION. 

2    4,6     7     20 
i  '  2     3     7     10 

1     i     375 
WHENCE,  2x2x3x7X5=420,  the  leail  common 
multiple  required. 

PROBLEM     II. 

fo  find  the  greatefl  common  meafure  of  two  or  more 

quantities* 

R  U  L  E. 

i.  FIND  the  greatefl  common  meafure-  of  any  two 
of  the  quantities,  by  dividing  the  greater  by  the  lefs, 
and  the  divifor  by  the  remainder  ;  and  fo  on,  divid 
ing  the  laft  divifor,  by  the  1  aft  remainder,  till  noth- 


(       So       ) 

ing  remains  •,  and  the  laft  divifor  made  ufe  of,  will  be 
the  greateft  common  meafure  of  thefe  two  quantities. 
2.  FIND  the  greateft  common  meafure  of  any  one 
of  the  other  quantities,  and  the  common  meafure  laft 
found ;  and  fo  on,  from  one  number  to  another,  thro' 
the  whole  j  and  the  laft  common  meafure  thus  found, 
will  be  the  greateft  common  meafure  required. 

EXAMPLES. 
Find  the  greateft  common  meafure  of  12  and 

OPERATION. 


3 

J  12 

HENCE,  3  is  the  greateft  common  meafure  required* 
Find  the  greateft  common  meafure  of  12, 18, 26, 36. 

OPERATION. 

F:rft  find  the    greateft  common  meafure  of  12 
and  i 8. 

thus, 


HENCE,  the  greateft  common  meafure  of  12  and 
18  is  6. 

Again,  find  the  greateft  common  meafure  of  6 
and  26, 

thus 


6(4 


Therefore  the  greateft  common  meafure  is  2. 

Laftly,  find  the  greateft  common  meafure  of  2 
and  36  : 


Confequently  the  greateft  common  meafure  of  1  2, 
1  8,  26,  and  36,  is  2;  which  was  to  be  done. 

PROBLEM     III. 

'fo  abbreviate,  or  reduce  a  Vulgar  Frattion  to  its  leajt 
or  moftjimple  terms, 

RULE. 

4f 

FIND  the  greateft  common  meafure  of  the  mime-* 
rator  and  denominator,  by  the  laft  problem  -,  then  di 
vide  them  by  their  greateft  common  meafure,  and  the 
refult  will  be  the  terms  of  the  fraction  required.  Or, 

DIVIDE  both  the  numerator  and  denominator  of 
the  given  fraction,  by  fuch  a  number,  as  will  divide 
them  without  a  remainder,  and  the  refulting  fraction 
in  the  fame  manner  5  and  fo  on,  till  no  number  ex 
cept  unity,  will  divide  both  without  a  remainder  -, 
and  you  will  have  the  fraction  required. 

EXAMPLES. 

Reduce  -^  to  its  moft  fimpla  terms, 

L  THE 


(     82     _) 

THE  grcateft  common  meafure  of  64  and  384,  is 
$4.     Therefore  64-7-64—1,  and  384-7-64—6  j  con- 

fequently  -TT-—  g>  the  fraction  required. 

Or,  Jl±i=l  ,  and  -1±*  -i;  /**/*«/  */  fo/w. 

'384-8     4B          4&->8     6' 

Find  the  value  of  ~,  in  its  moft  fimplc  terms, 

Thus.  -  ^"^iz—,  the  fraflion  required. 
45-^5     9 


Reduce  121,  to  its  moft  fimple  terms.         /Inf.  - 
480  5 

PROBLEM    IV. 

ff"0  wn/^1  a  mixed  number,  in  the  form  of  a  Vulgar 
Fraftion. 

RULE. 

MULTIPLY  the  whole  number  with  the  denomi 
nator  of  the  fraction,  and  to  the  product  add  its  nu 
merator  ;  then  under  this,  write  the  faid  denomina 
tor  j  and  you  will  have  the  fraction  required, 

EXAMPLES. 

Write  4J,  in  the  form  of  a  Vulgar  Fraction. 
Thus,  4X2—8,  and  8  +  1=9  /^  numerator-, 

Whence  ~  is  the  fraction  required. 


„  6  _  126  ,  n  i      20      40X100+20 

aT-g.—  .  i      zz"       >  ana  40.  .    •—  •   •.. 

10  100  100 


100  •     2C  2e  20 

PROB- 


PROBLEM    V. 

fo  find  the  'value  of  an  improper  fr  a  ftion. 

RULE. 

DIVIDE  the  numerator  of  the  given  fradtion  by 
the  denominator  5  and  the  quotient  will  be  the 
value  fought. 

EXAMPLES. 
Find  the  value  of  —  2L. 

12 


20         417 

-M  00=40  -  j    —  =20~£, 
j;oo      ao 

PROBLEM    VI. 

Ti  write  a  whole  number  in  the  form  of  &  Vulgar 
Fraction)  whoje  denominator  is  given. 

RULE. 

MULTIPLY  the  whole  number  with  the  given  de- 
aominator  j  and  under  this  produft  write  the  faid  de 
nominator;  and  you  will  have  the  fraction  required. 

EXAMPLES. 

Reduce  40  to  its  equivalent  Vulgar  Fraftion^ 
whofe  denominator  is  10. 


Thus, 
Whence,  42—  is  thefraRion  required* 

Change  304  into  its  equivalent  Vulgar  Fraftion, 
having  5  for  its  denominator, 

Thus, 


Thus,  =         the  fraction  required. 

Change  3476  into  its  equvialent  Vulgar  Fraction, 
having  12  for  its  denominator. 


Thus,  n  the  f  ration  required. 

PROBLEM     VII. 

f  o  alter  or  change  a  Vulgar  Fraction  into  another  of 
equivalent  value  -,  wbofe  denominator  is  given. 

RULE. 

MULTIPLY  the  given  numerator  with  the  prbpofed 
denominator  ;  the  product  divided  by  the  denomi 
nator  of  the  given  fradtion,  will  give  a  new  numera 
tor  ;  under  which  write  the  propofed  denominator  ; 
and  you  will  have  the  fraction  required. 

EXAMPLES. 

Change  —  into  its  equivalent  Vulgar  Fraction, 
v/hofe  denominator  is  20. 

Thus,  -  -  =10  the  new  numerator. 

Therefore,  —  is  the  fraction  required. 

Change  ~*  into  its  equivalent  Vulgar  Fraction, 
having  40  for  its  denominator. 


Thus,  "*    ^-izjo  :  therefore  -~°  is  the  fraftion  req. 


Change  ~  into  its  equivalent  Vulgar  Fraction, 

whofe  denominator  is  24. 

Thus, 


req. 

20  2.0 

PROBLEM    VIII. 

fo  change  a  Vulgar  Fraction  into  another  of  equiva- 
lent  value,  whofe  numerator  is  given. 

RULE. 

MULTIPLY  the  given  denominator  with  the  pro- 
pofed  numerator  -,  and  the  produft  divided  by  the 
numerator  of  the  given  fraftion,  will  give  a  new  deno 
minator  5  over  which  write  the  propofed  numerator  ; 
and  you  will  have  the  fra&ion  required. 

EXAMPLES. 

Change  —  into  its  equivalent  Vulgar  Fraction, 
whofe  numerator  is  20. 
Thus,  —  ^  —  —40  :  therefore,  —  ,  is  the  fraftion  req. 

7     ...  .$ 

Change  —   into  its   equivalent  Vulgar  Fra&ion, 

whofe  numerator  is  8. 


Thus,          =IO|  .  therefore,  —  p  w  thefraffion  req. 

1  IOT 

Change  -  •  into  its  equivalent  Vulgar  Fraftion, 
whofe  numerator  is  37.  Anf.  2~~ 

PROBLEM    IX. 

To  reduce  a  mixed  fraftion  to  fimfle  terms. 

RULE. 

^  i.  REDUCE  the  numerator  and  denominator  of  the 
given  fraftion  to  improper  fradions. 

i,  MULTIPLY 


(      86      ) 

2.  MULTIPLY  the  numerator  of  the  denominator, 
into  the  denominator  of  the  numerator,  for  a  new  de 
nominate*  j  and  multiply  the  numerator  of  the  nu* 
merator,  into  the  denominator  of  the  denominator, 
for  a  new  numerator  ;  and  you  will  have  the  terms  of 
the  fraction  required. 

EXAMPLES. 

4.— 

Reduce  ±2  to  fimple  terms. 

?T 
A.  *  *  ? 

Firft,  If— (by  reducing  to  impr.  fract.)  JL  iz 

fi 

~-$Q  the  f raft  ion  required, 

OQ 

O  1 

Reduce  ~J  to  (imple  terms. 
10 

Thus,!!-^-^!^!?.  andi|i=:^r:^. 
10     10     2X10     20          16       16     48 

10 


_20O^    3OO     __3OO      _ 


PROBLEM    ,X. 

3*o  reduce  a  compound  fraffion  tq  a  Jimfle  one  of  c~ 

qua!  value* 

RULE. 

i.  REDUCE  all  fuch  parts  of  the  given  fraction  as 
are  whole  numbers,  mixed  numbers,  and  mixed  frac 
tions;  according  to  the  foregoing  rules  j  that  is,  whole 
and  mixed  numbers  muft  be  reduced  to  improper 
fractions^  and  mixed  fractions  to  fimple  terms. 

a.  Mui,* 


2.  MULTIPLY  all  the  numerators  continually  to 
gether,  for  a  new  numerator,  and  all  the  denomina 
tors  continually  together,  for  a  new  denominator  ; 
and  the  former  produd  written  above  the  latter*  will 
give  the  fra&ion  required. 

Note.     Any  number  that  is  found  among  the  nume 
rators  and  denominators,  may  beftruck  out  cfboth, 

EXAMPLES. 

Reduce  5  of  lof-^,  to  a  fimple  fraftion. 
p  3       4.6 

Fhus, 2X3X*=(by  ftriking  out  the  3)^!=  ~ the 

3X4X6  J'4X6     24 

raftion  required. 

O  *"7 

Reduce  ±  of  /- ,  to  a  fimple  fraftion-* 
4      9r 


—  ~  ; 
19 

f-X  19=76  /^<?  »^w  denominator  :  therefore 


,L.  —  ~  ;  then  3  x  1  4=4^  the  new  numerator,  and 
9r      19 


76 
raftion  required. 


^flofS^Ioffoflof^i^. 
264*-  269      i     108 

toff  of -E=!  of  5  of  l!=l^><ii-=J£L. 
3      5      V     3      5      88     3X5X^8     1320 

PROBLEM     XL 

f<?  r^r^  feveral  fractions  of  different  denominators, 
to  equivalent  fraffionsybaving  a  common  denominator. 

RULE. 
i,  REDUCE  all  fraftions  to  fimple  tqrms. 


(       38       ) 

2.  MULTIPLY  each  numerator  into  all  the  deno 
minators  except  its  own,  for  new  numerators. 

3.  MULTIPLY  all  the  denominators  continually  to-  1 
gether,  for  a  new  and  common  denominator,  and  this  i 
written  under  the  feveral  new  numerators,  will  give1 
the  fractions  required. 

EXAMPLES. 

Reduce  -*  -,  and  ^,  to  their  equivalent  fractions, 
having  a  common  denominator. 

f  IX4X6=24  the  new  numerator  for  \. 

Firft,  <  2x3X^=36  the  new  numerator  for  |. 

(.  5X2X4=40  the  new  numerator  for  4. 

Then  2x4x6=48  the  new  and  common  denominator, 

Hence  -A   ~,  and  ~,  are  the  fractions  required. 
40     4^          4* 

~,  -,  and  -,  reduced  to  a  common  denominator-^    ^    \ 
9  7X4X9* 


1X7X4^144  and 

^^'' 


7X4X9          7X4X9^252^252'         252" 

-  and  -  of  ~of  ~,  reduced  to  a  common  denominators* 
3        3      5      7r 

22,  and  3X4Q8  =.i3£Q> 


£224. 
3X13^          3X1320     3960          3960 

3?r)  «J,  and  -  of  4,  reduted  to  a  common  denominators 
4  3  , 

720   189        ,  448 
'     ,  _|,  and  S-, 

336   336          336 


PROB- 


PROBLEM    XII. 

To  reduce  feveral  fractions  of  different  denominators,  to 
gthers  of  equivalent  value,  having  the  leaft  foj/ible 
common  denominator* 

R  U  L  E. 

1.  REDUCE  all  the  fractions  to  fimple  terms. 

2.  FIND  the  leaft  common  multiple  of  all  the  de 
nominators  j  and  you  will  have  the  leaft  common  de 
nominator  required. 

3.  DIVIDE  the  denominator  thus  found  by  the  de 
nominator  of  each  fraction,  and  multiply  the  quo 
tient  with  its  numerator,  and  you  will  have  new  nu 
merators,  under  which  write  the  common  denomina 
tor  ;  and  you  will  have  the  fractions  required. 

EXAMPLES. 

Reduce  ~,    -,  and  -  to  equivalent  fractions,  that 
84          2 

flull  have  the  leaft  poflible  common  denominator. 
Firft,  the  leaft  common  multiple  of  8,  4,  and  2,  is  3  ,: 
Then,  8  —  8  x  i=J>  the  new  numerator  for  ^ 
And,  8-f-4X3==^  *be  new  numerator  for— 
Alfo,  8-7-2XI:==4>  the  new  numerator  for  ~. 
Hence  the  fractions  required  are  4*  TJ  an<i  •?• 

Reduce  i,    -,    ~,  and  ~,  to  equivalent  fractions, 

I     4     5  " 

having  the  leaft  poflible  common  denominator. 

.     Firft,  the  leaft  common  multiple  of  the  3,  4,  5, 
and  6,  is  60. 
Then,  60-^3X1=20,  the  new  numerator  for  \- 

>nd,  6o-r-4Xj::=4S,  tie  new  numerator  for  ~ 

M  Alfo. 


(      9°      ) 

A!ib,~6o-f-  5  X4~4-S  tie  new  numerator  for  -J 
Laftly,  6o-~6x5~50  the  new  numerator  for 


Hence,  ~,       ,      -,  and      ,  <w  the  fraftions  red. 

OO       DO      OO  rx  6O 

P-RQBLEM     XIII. 

fo  change  the  fraction  of  one  denomination  to  thefraftion 
of  a  greater  one>  retaining  its  fame  value. 

RULE. 

CHANGE  the  given  fraction  into  a  compound  one, 
by  writing  its  value  in  all  the  intermediate  denomi 
nations  up  to  the  one  wherein  the  value  of  the  frac 
tion  is  to  be  ex;)refied  ;  and  the  value  of  this  com 
pound  fraftion,  will  be  the  fraction  required, 

EXAMPLES. 
Change  -  of  a  nail,  to  the  fraction  of  an  ell  Eng, 

Firft,  .»•  of  a  nail~=.~  of  a  quarter,  and  -rr:-  of  an  ell  : 
3  3  4    5r 

Therefore,  -  of  a  nail^z-  of  '-  of  -—-£->  the  fraction  req. 
3      4      5*    oo 

1  pennyweights,  reduced  to  the  fraction  of  a  pound 
=r  —  of  —  •=  —  .      3  grains,  reduced  to  fraction  of  an 

2fr          IX      240 

ounce=:—  of  —  •=—  |—.         of  a  cent,  reduced  to  the 
24      ao    480       3 

fraction  of  am  ilree  of  Portugal-  of-  -  =  -  ;     10 

3       124      j/a 

cents,  reduced  to  the  fraction  of  a  pound  flerling  of 

Ireland^:-    ••=  J-4     ~  of  a  cent,  reduced  to  the  frac- 
410    41       8 

tion 


*7  T  *7 

rion  of  a  dollar-^  of =o^-.      i  drachm  Avoir- 

o       i oo    8  oo 

i     r  i     ~    x      f  r     f 

is=--i  of  -7  of of  —  of  a  tun. 

10         10         112         2O 

PROBLEM     XIV. 

2V  change  the  fraRion  of  one  denomination  to  the  frac 
tion  of  a  lefs  one,  retaining  itsjame  'value. 

RULE. 

MULTIPLY  the  numerator  of  the  given  fra&ion 
into  all  the  intermediate  denominations  down  to  the 
one  wherein  the  value  of  the  given  fradtion  is  to  be 
exprefled,  and  under  this  product,  write  the  given  de 
nominator,  and  you  will  have  the  fra&ion  required. 

EXAMPLES. 

Reduce  —  of  an  ell  Eng.  to  the  fraction  of  a  nail. 
70 

Thus,  1X5X4=20  the  numerator 
Therefore,  — =~/V  thefrattion  required. 

Reduce  — ^-  of  a  lb  Troy  to  the  fraft.  of  a  grain. 


Thus,  ::=  .f  btffafKm 

II2O  1120 

—  of  a  pound  Troy,  reduced  to  the  fraction  of  a 


.  i       aXiaXao      480          8         c       , 
pennyweight—  --  —  -  —  ;  -  of  an  hun- 
1240         1240     17920 

dred  weight,  reduced  to  the  fra&ion  of  an  ounce= 

8XH2- 


J_  of  a  milree  of  Portugal, 

17910  J7920  372 

reduced  to  the  fra&ion  of  a  cent=—  1^——  =-. 

$7*       57*.    5 

PROBLEM     XV. 

To  find  the  value  of  a  Vulgar  Fr  a  ft  ion  in  known  farts 
of  the  integer. 

RULE. 

MULTIPLY  the  numerator  of  the  given  fra&ion 
with  the  parts  in  the  next  inferiour  denomination,  and 
divide  the  produA  by  the  denominator  $  then  if  there 
^e  any-  remainder,  multiply  it  with  the  parts  in  the 
next  inferiour  denomination,  and  divide  by  the  former 
divifor,  and  fo  on,  and  the  feveral  quotients  refulting 
v/ill  exhibit  the  value  fought, 

EXAMPLES. 

Find  the  value  of—  of  an  ounce  Trov, 
24 

OPERATION, 

5 

£0 

24  \  100(4  pennyweights* 
'  96 

4 
24 


Therefore* 


(      93      ) 

* 

Therefore,  -*•  of  an  ounce=4  dwt.  4j?r.  /J 
M 

fought. 

Find  tfic  value  of  -  of  an  ounce  Troy. 

OPERATION. 
5 

20 

7\ioo( 
/   14     irtm. 

—  24 

7)48( 
f  6     6rtm. 

Therefore  i^dwt.  66rgr.  is  the  valug  fought. 

Find  the  value  of  -  of  an  hundred  weight, 
7 

OPERATION. 
6 
4 


3 
—  28 


Therefore  3qr.  ia/^.  is  tbe  value  fought. 

Find  the  value  of  —  of  a  pound  fieri,  of  Ireland 
4< 

Thus, 


(      94      ) 


us>     -—  jo  cts.  the  valuffougbf. 
4* 

Find  the  value  of  —  of  a  pagoda  of  India. 
Thus,  a    I94^=4^.  /fo  value  fought. 

PROBLEM     XVL 

21?  reduce  the  known  farts  of  an  integer  to  their  equiva- 
lent  Vulgar  Fraction. 

RULE. 

1.  REDUCE  the  given  parts  to  the  lead  denomina 
tion  mentioned. 

2.  REDUCE  the  integer  to  the  fame  denomination  ; 
and  the  latter  written  beneath  the  former,  will  be  the 
fra&ion  required, 

EXAMPLES. 
Reduce  3  dwt.  *]gr.  to  the  fra&ion  of  a  pound.  > 

OPERATION. 
dwt.  gr.  cz. 

37  12 

24  20 

79  240 

—  24 

960 

480 

5760 
Therefore,  -^-  is  thtfraftion  required. 

Reduce 


C      95      ) 

Reduce  10  cts.  to  the  fra&ion  of  a  potand  fterling 
of  Ireland. 

Thus,  —  is  tfa  fraction  required. 


n.  reduced  to  the  fraction  of  a  foot— T-f^JZ. . 

12        10 

4-  p.  reduced  to  the  fra<ft.  of  an  acre=~:rr--rnr-— . 

1 60    480    30 


CHAP.      III. 

ADDITION,  SUBTRACTION,  MUL 
TIPLICATION,  A*D  DIVISION  OP 
VULGAR  FRACTIONS. 

SECT.       L 

Of  ADDITION  of  VULGAR  FRACTIONS. 

RULE. 

EPUCE  all  the  fractions  to  a  common  de- 
nominator,  by  the  rule  to  problem  xi  of  the 
.aft  chapter  :  thofe-  of  different  denominations  to  the 
Tame,  by  the  rules  to  problem  xm  or  xiv. 

2.  ADD  all  the  numerators  together  for  a  new  nu 
merator,  under  which  write  the  common  denomina- 
:or  ;  and  you  will  have  a  fraction  equal  to  the  fum 
*e  quired. 

EXAMPLES, 

the  fum  of- — | — }-- . 

ZT|T5| 

Thus, 


(      96      ) 


Thus,  -+--)--  =(by  reducing  to  a  common  derio- 


mirutor)  i?+     +     = 


24         24  24  24 

2  *          I 

Required  the  fum  of  2-H  —  ?+*  of  4. 

4       o 

,24"^M-^  of  4=  j+^+|= 
720     189 


336       36.336~  336 

Find  the  fum  of  -  of  a  grain  -^  —  of  an  ounce. 

•7  7  I  I  ? 

Firft,  -0/<z  grain—-  of  —  of  —  n-*—  ^/^«  ounce  •, 
*  4  4       24       20      1920  y 

/ir/;  /£<?  /^^  becomes  -  4-*^:  -2  -  tbt  ftim  req. 
1920  '  7     13440      y 

SECT.      II. 
Of  SUBTRACTION  of  VULGAR  FRACTIONS. 

RULE. 

1.  PREPARE  the  fra&ions  as  in  Addition. 

2.  SUBTRACT  the  numerator  of  one  fraction  from 
the  numerator  of  the  other,  and  the  refult  placed  a- 
bove  the  common  denominator  will  be  the  differ 
ence  required. 

EXAMPLES. 

From  -take  ~. 
3          4 

Thus,  ~~-=s±£.~-^  ~iH^—  J.  fa  difference  re^f 
3    4     ia     12      12       ia 

From 


4  l       r    l 

From  -  take   -of  -. 

5  3       3 

~,       4     i    cl     4     i     36      5_36~5     31 

THUS,  -_ -  Of  r-- -=- -~  -  :—  =  -  ,* 

difference  required. 

3      i      2__io       6    _io8o      i8__io8o— 18 
~°  9°     "~"~"""          "~"" 


_io62  y     5     51  _44O     204 236 

From  -  of  an  ounce  take  -  of  a  grain 

7  7 

Firft,  -  of  a  grain=. — —  of  and  ounce  :  Therefore, 

5  3  9589  j.jr 

-. = .  ts  jfo  difference  required. 

7     1920     13440 

SECT.      III. 
Of  MULTIPLICATION  of  VULGAR  FRACTIONS, 

RULE. 

1.  REDUCE  all  whole  and  mixed  numbers  to  im 
proper  fractions,  mixed  fractions  to  fimple  terms,  and 
fractions  of  different  denominations  to  the  fame. 

2.  MULTIPLY  all  the  numerators  together  for  ft 
new  numerator,  and  all  the  denominators  together 
for  a  new  denominator;  and  you  will  have  the  terms 

"the  fra&ion  required. 

EXAMPLES. 

I        o 

Required  the  product  of  ~X  ~- 

\5  T" 


3 

'hus,  TTT^—^  thefroduft  required. 


N  64X 


(     98       ) 


4          ^4-  v  12       IO       I2XIO       12O 

~  X~  —  (by  reduction)  —  X—  =  —  r^  —  =  - 
34.4  '  10     12     10X12     120 


J    ~- 

;         4     "^3 

SECT.       IV. 

Of  DIVISION  of  VULGAR  FRACTIONS. 

RULE. 

PREPARE  the  numbers  as  in  Addition,  then  multi 
ply  the  numerator  of  the  divifor  into  the  denomina 
tor  of  the  dividend,  and  the  numerator  of  the  divi 
dend  into  the  denominator  of  the  <3ivifor  5  then  the 
latter  written  above  the  former,  will  give  the  quo 
tient  required.  Or, 

INVERT  the  divifor,  that  is,  write  the  denomina 
tor  in  the  place  of  the  numerator,  and  the  numera 
tor  in  the  place  of  the  denominator  ;  then  proceed  as 
in  Multiplication,  and  the  refult  will  give  the  quo 
tient  required. 

EXAMPLES. 

Required  the  quotient  of  2-r-,« 
Thus,  1X4—  4-thenumeraton  and  i  writhe  denm. 


4 
Therefore,  -±=2  is  the  quotient  required. 

414 
Or,  -  X""—-  the  fame  as  before. 

i       22 

2       8  _27      2_2-X2_54     4      4_J>8      4 
3~:  27"""  8  X3~  SX3  ""24;  7  '  is"  4  X7" 

28X4 


99) 


=II^=45  lM^=(by  reduction)-*  -J  = 
28  8       i  '2       2 


4X7       28  i  24 

4X2       8       i      i  m  i    f        i  .  i__3Xi-3 
'  4      2*3       '-ST'iXS-S- 


Mifcellaneous  £>ueftions. 

AM  AN  at  hazard  won  the  firft  throw  2|  dol 
lars — the  fecond  throw  he  won  as  much  as  he 
then  had  in  his  pocket — the  third  throw  he  won  4 
dollars,  and  the  fourth  throw  he  won  double  of  all 
that  he  then  had,  at  which  time  he  found  that  he  hacj 
in  all  45  dollars.     How  many  had  he  at  firft. 
Anfwer.     3  dollars. 

THERE  is  a  certain  club,  whereof^-  are  merchants, 
4-  mathematicians,  ~  mechanics,  and  13  phyficians. 
How  many  were  there  in  the  whole  ? 

Anfwer.     60. 

REQUIRED  the  difference  between  three  times  thir 
ty-three  and  a  third  j  and  three  times  three  and  thir 
ty  and  a  third. 

Anfwer.     6oJ. 

A  MAN  who  was  driving  fome  flieep  to  market, 
was  met  by  another  who  demanded  the  number  of 
Iheep  in  his  drove  :  the  drover  to  evade  a  direcl:  anf- 
wer  replies,  that  if  I  had  as  many  more,  and  half  as 
many  more,  and  12!  fheep,  I  fhould  have  100. 
What  number  had  he  ? 

35- 

PART 


Anfwer. 

\ 


PART      III. 

CONTAINING    THE    DOCTRINE    OF 
DECIMAL    FRACTIONS. 


C  H  A  P.      I. 

¥ 

DEFINITIONS  and  ILLUSTRATIONS. 

AD  E  C  I  M  A  L  Fraction  is  formed  from  a 
proper  Vulgar  Fraction,  by  dividing  the  nu 
merator  with  cyphers  annexed  to  it,  by  the  denomi 
nator  ;  that  is,  the  equivalent  Decimal  of  any  Vulgar 
Fraction  is  found  by  multiplying  the  numerator  with 
10,  100,  or  1000,  &c,  till  it  be  fo  increafed,  that  it 
rnay  be  exactly  meafured  by  its  denominator  j  and 
this  quotient  will  be  the  decimal  required  : 

Thus,  ~  X  i oo— ^21122  — 122--; 25  j  and  »Xio  ~ 
4  44  2 


22  4  44 

which  quotients  are  exprefTed  by  writing  them  with 
a  point  on  the  left-hand  :  Thus,  ^=:  .25,  i  =  -5>  and 
J  =  -75  i  which  are  reflectively  equal  to  TVoj  ~,  and 
.jVo-s  but  thefe  denominators  are  always  omitted,  and 
the  numerators  written  as  above,  where  the  point  dif- 
tinguifhes  them  from  whole  numbers  :  Thus,  2.3=; 
&c, 

HENCE 


HENCE  it  appears  that  every  Decimal  Fraftion,  is 
equal  to  a  Vulgar  one,  whofe  numerator  is  the  deci 
mal,  and  the  denominator  unity,  with  as  many  cy 
phers  annexed  to  it  as  there  are  places  of  figures  in 
the  numerator :  Thus,  .1,  .44,  and  .127,  are  refpec- 
tively  equal  to  TV,  -rVo*  and  -r^J-J-- 

THEREFORE  it  follows,  that  in  decimals,  unity  is 
divided  in  10,  100,  or  1000,  &c.  equal  parts;  and  the 
given  decimal  reprefents  the  number  of  thofe  parts  : 
Thus,  .  i  =  _~  reprefents  one  tenth  part  of  an  unit, 
.44  reprefents  forty-four  hundred  parts  of  an  unit, 
&c.  Therefore,  in  decimals,  cyphers  annexed  neither 
increafe  nor  diminilh  their  value  ;  but  cyphers  pre 
fixed,  diminifh  their  value  in  a  ten  fold  proportions 
Thus,  .440— x\V6=(by  £ne  nature  of  Divifion)  Tt0*j 
rz.44  ;  but  .04— T-O-O~TO-  °f  (Ar)  -4>  an^  f°  °n  for 
any  other  decimal. 

WHENCE  it  follows,  that  the  farther  any  diget  or 
numeral  figure  (lands  from  the  units'  place,  or  deci 
mal  point  towards  the  right-hand,  the  lefs  will  be  its 
value,  to  wit,  in  a  tenfold  proportion.  Thus  in  the 
decimal  .mi,  the  figure  next  to  the  decimal  point 
is  ^-V>  tne  fecond  is  -y-^,  the  third  -j-oVerj  and  the 

T_.L^_,  which  is  plainly  a  feries  of  numbers  in  ge 
ometrical  proportion,  decreafing  by  the  common 
divifor  10.  Again  ,oi23zn:T4-c-f-T-51oo+T^4-^-i  an<^ 
the  like  to  be  underftood  of  all  others. 

HENCE,  the  notation  of  decimals,  or  the  valuation 
of  the  feveral  places  from  unity  downwards,  is  the 
fame  among  themfelves  as  that  of  integers  or  whole 
numbers  *,  therefore  every  figure  is  to  be  valued  accord 
ing  to  the  diftance  it  Hands  from  unity  downwards. 


CHAP, 


CHAP.      II. 

ADDITION,  SUBTRACTION,  MUL 
TIPLICATION,  AND  DIVISION  OF 
DECIMAL  FRACTIONS. 

SECT.       I. 

Of  ADDITION  of  DECIMALS^ 
RULE. 

i.  TTT  7  R  I  T  E  the  given  decimals  in  fuch  order, 
VV     that  thofe  places   of  equal  diftance  from 
unity  or  the  decimal  point,  may  Hand  directly  un 
der  each  other. 

a.  Find  their  fum  as  in  whole  number?,  then  dif- 
tinguifh  with  a  point  as  many  places  of  figures  on 
the  right-hand,  as  are  equal  to  the  greateft  number 
found  in  any  given  decimal  $  and  you  will  have  the 
fum  required. 

EXAMPLES. 

Find  the  fum  of  .176  +  .1264+.  34+^94 

Thefe  numbers  being  placed  according    to  the  rule 
will  Hand 

.i76 

4 


Find   the  fum  of    34.  123+6437,  27+34?  •£ 

^347^34+347^34-1. 

thus, 


r         34-123 

thus         6437'27 
thUS'          347-2 


347634.1 


354454.040634=1/^1  required. 


Required  the  fum  of  25.1244-12,247  +  24.3485 
25  124 


4578-74 

4992.5595— /#>#  required. 


SECT.      II. 

Of  SUBTRACTION  -of  DECIMALS. 

RULE. 

WRITE  down  the  numbers  as  in  Addition,  then 
fubtraft  the  lefs  from  the  greater  as  in  whole  num- 
bersj  remembering  to  point  off  in  the  remainder  as  in 
Addition  j  and  you  will  have  the  difference  fought. 

EXAMPLES. 
Required  the  difference  between  12.19,  and  8.9 


3  .  2  9  r:  dffiren  ce  required. 

Required 


104      ) 

Required   the  difference   between  342.364,  and 
-199.2437 

thus  1342.364 

us>{  299.243? 


43.  i  v&yssjffirince  required* 


—  1 999-99998:=473-00242  > 

2479-3777—93o.oooo45=i549.377 
9999.8888—8888.9999—1 1 10.8889 

SECT.       III. 

Of  MULTIPLICATION  of  DECIMALS. 
RULE. 

WRITE  the  numbers  and  multiply  them  as  in  com 
mon  Multiplication  ;  then  diftinguiih  with  a  point 
as  many  places  of  decimals  in  the  product,  as  are 
equal  to  the  number  in  both  factors  $  and  you  will 
have  the  product  required. 

ftfote.  If  the  number  of  places  in  theproduff,  are  kjs 
than  the  number  of  decimal  places  in  both  faftors,, 
you  muftfiipply  the  deficiency  by  prefixing  cyphers. 

THAT  the  number  of  decimal  places  in  the  prod 
uct,  ought  to  be  equal  to  the  number  in  both  factors, 
may  be  thus  demonftrated. 

SUPPOSE  .34  were  to  be  multiplied  with  .27  j  the 
product  of  theie  two  numbers  by  common  Multipli 
cation  is  918  ;  but  .34==T35*S.  and  .27=TVg- ;  there 
fore,  .34  X  .27  =  £&  X  -rVo  =  -1-1-44-5-=  (by  thena- 
ture  of  decimal  notation)  . 091 8,  confiding  of  as  maray 
places  of  figures  as  there  were  in  both  factors  j  and 
the  fame  will  hold  true  in  any  others.  ^.  E.  D. 

E  XAM- 


EXAMPLES. 

Required  the  produdt  of  2.438  ^  ,005. 
OPERATION. 

2.438 
.005 

.o  1 2 1 90  ^=.produft  required. 
Required  the  produft  of  34.38X24,7 
OPERATION. 

34.38 
24.7 

24066 
6876 

required. 


Required  the  product 


3. 8  402  z:  produfi  required, 
Required  the  produft  of  2.7122X3,2121 


64242 
64242 
32121 

2*4847 
64242 


IN  the  multiplication  of  decimals,  where  the  factors 
confift  of  a  great  number  of  decimal  places,  the  oper 
ation  becomes  very  prolix,  and  befides,  a  great  part 
of  it  is  entirely  ufelefs,  fmce  that  four  or  five  places 
of  decimals  in  the  product,  is  fufficient  for  common 
purpofes.  Therefore  to  abridge  the  work  by  obtain 
ing  the  product  true  to  any  defigned  number  of 
places  of  decimals,  you  muft  obferve  the  following 

RULE. 

1 .  WRITE  the  multiplier  inverted,  fo  that  the  units' 
place  mayftand  under  that  figure  of  the  multiplicand, 
to  whofe  place  the  product  is  to  be  found  true. 

2.  IN  multiplying  with  the  feveral  figures  of  the 
multiplier,  you  muft  reject  all  the  figures  of  the  mul 
tiplicand,  that  are  to  the   right-hand  of  the  figure 
you  are  multiplying  with  ;  placing  the  firft  figure  of 
the  feveral  products  directly  under  each  other,   in- 
creaied  by  adding  I   from  5  to  15,  2  from  15  to  25, 
&c.  of  the  product  of  the  multiplying  figure  with 
the  proceeding  figure  of  the  multiplicand,  when  you 
begin  to  multiply  5  and  the  fum  of  all  the  products 
will  be  the  product  required. 

EXAMPLES. 

Required  the  product  of  3.2121x2.712,  to  three 
places  of  decimals. 


3.2121 


0/3.212X2  . 

0/3.21X7,  increafed  by  adding  i  for 
0/3.2x1  [the  f  rod.  0/7x2 

'/3X2 

required* 

Required  the  product  ©f  3.24211X2.34634,   to 
four  places  of  decimals. 

3.24211 
436432 

64842=3.2421X2 

9726=3.242X3. 

1 297 =3. 24x4*  increafed  by  adding  i  far  4X2 

194=3. 2X  6>  increafed  by  adding  2  for  6X4 

™~3X3>  increafed  by  adding  i  for  3x2 


required. 

Required  the  produd  of  2,13214X2.21 134,    to 
five  places  of  decimals. 

2.13214 
431122 


426428=2.13114X2 

42643=2.1321X2,  increajedby  adding  ifor  2X4 
21.32=2.132X1 
213=2.13x1 

64=2. 1X3,  increafed  by  adding  i  for  3X3 
2=2X4 


required. 

Required 


Required  the  product  of  27.i7X*9»i4>  in  inte 
gers  only. 
27.17 
4191 

27  2  =  27  .  1  X  i  >  increafed  by  adding  I  for  I  x  7 
244—  27  X9>  increafed  by  adding  i  /0r  9X1 
increa/ed  by  adding  i  /<?r  1X7 
increa/ed  by  adding  I  /0r  4X2 


required. 

SECT.       IV. 

Of  DIVISION  of  DECIMALS. 

IN  divificn  of  decimals,  it  may  at  firft  appear 
difficult  to  determine  the  number  of  decimal  places 
the  quotient  muft  coniift  of  ;  but  this  difficulty  will 
vanifh,  when  we  confider  that  the  quotient  mufr  be 
fuch  a  number  that  when  multiplied  with  the  divi- 
for  will  produce  the  .dividend;  therefore  it  follows, 
that  the  number  of  decimal  places  in  the  divifor  and 
quotient  taken  together,  muft  be  equal  to  the  num 
ber  in  the  dividend,  by  the  nature  of  Multiplication  ; 
confequently  the  difference  between  thofe  in  the  di 
vifor  and  dividend,  muft  be  equal  to  the  number  in 
the  quotient  j  which  affords  the  following 

'  R  U  L  E. 

RANGE  the  numbers  and  divide  them  as  in  com 
mon  Divifion,  then  point  off  as  many  places  of  de 
cimals  in  the  quotient,  as  are  equal  to  the  difference. 
between  thofe  in  the  divifor  and  dividend  ;  and  you 
will  have  the  quotient  required. 

Note 


Note  i.     If  there  are  not  fo  many  places  of  figures 
in  the  quotient,  as   are  equal  to  the  difference  be 
tween  thofe  in  the  divifor  and  dividend,  you  muft 
Jupply  the  defett  by  prefixing  cyphers. 

2.  If  the  places  of  figures  in  the  dividend,  are  lefs 
in  number  than  thofe  in  the  divifor,  you  muft  an 
nex  cyphers  to  the  dividend. 

EXAMPLES. 
Required  the  quotient  of  849.186  divided  by  24.7 

OPERATION. 

24.7  ^849.1 86(34.38 -^quotient  required. 
'741 

1081 

988 

MM^BM 

938 

741 


1976 
1976 


Note.  If  the  dhifor  be  10,  or  100,  &V.  the" quotient 
may  be  found  by  removing  the  decimal  point  in  the 
dividend ,  as  many  places  towards  the  left-hand 
as  there  are  cyphers  in  the  divijor  :  thus,  the  quo 
tient  of  1000)2737,45  is  2.73745  and  .0234-7- 

IOO—.000234. 

Required  the  quotient  of  .012190-7-2.438 

OPER- 


(      no     ) 

OPERATION. 

2.438 \. 012190(5 
)    12190 


VJ 

HERE,  the  quotient  found  by  divifion  is  5  ;  but 
the  difference  between  the  decimal  places  in  the  di- 
vifor  and  dividend  are  three  j  therefore  .005  is  the 
quotient  required. 

Required  the  quotient  of  2-~42. 
OPERATION. 

42X200000(^04761  &c.— quotient  required. 
A68 

320 
294 

260 

252 

80 
42 

38  &c. 

Required  the  quotient  of  1 65, 6995001 2964- 


OPER- 


(  I"  ) 

OPERATION. 

52.7438  ^165.6995001 296(3, 141  sw 
J  1582314 


746810 


2193720 
2109752 

839681 


3122432 
2637190 

4852429 
4746942 

1054876 
1054876 

o 

HERE,  as  in  Multiplication,  the  work  may  be 
greatly  contra&ed,  by  finding  the  quotient  true  to 
any  determinate  number  of  decimal  places  :  The 
method  is  as  follows. 

RULE. 

i.  RANGE  the  numbers  -as   in  common  Divifion. 

a.  TAKE  the  figures  of  the  given  divifor,  to  as 
many  places  of  decimals  as  you  intend  the  quotient 
fhall  confift  of,  for  your  firft  divifor,  and  find  a  quo 
tient  figure  by  comparing  this  divifor  as  in  common 

Divifion  ; 


Divifion  ;  then  fubtraft  its  product  with  the  divifor, 
from  the  dividend  as  ufual,  calling  the  remainder  a 
new  dividend. 

3.  REJECT  the  right  hand  figure  of  your  former 
divnfor,  and  call  the  refult  a  new  divifor  ;  then  find 
a  quotient  figure  by  comparing  the  new  divifor  and 
dividend  together,  and  place  it  in  the  former  quo 
tient,  fub  trading  as  before  j  and  fo  on,  making  each 
remainder  a  new  dividend,  and  rejecting  the  right- 
hand  figure  of  the  laft  divifor  for  a  new  one  j  alfo 
remembering  to  add  for  the  figures  rejected  as  in 
Multiplication. 

Note  i .  If  there  are  notfo  many  places  of  decimals 
in  the  divifor,  as  you  intend  there  foall  be  in  the 
quotientljupply  the  defeft  by  annexing  cyphers. 

2.  Tou  may  determine  hew  many  places  of  whole 
numbers  there  will  be  in  the  quotient,  by  confider* 
ing  that  the  fir  ft  figure  of  the  quotient y  is  always 
of  the  fame  denomination  of  that  figure  of  the  divi* 
dendy  which  ft  ands  direftly  over  the  units'  place  of 
the  produft  of  the  fir  ft  quotient  figure  and  divifor. 

EXAMPLES. 

Required  the  quotient  of  10.1 934-7-4. 2,  to  three 
places  of  decimals. 


QPE& 


(      "3      ) 

required, 


J  8400 

420)1793 
1680 

42)113 
84 

4)29 
28 

i 

Required  the   quotient  of  165.6995001296-7- 
52.7438,  to  five  places  of  decimals. 

52.74380  ^165.6995001  296(3.  141  592=  quotient  req. 
J  15823140 


52.7438)746810 
5274JS 

52.743)219372 

2i0975=52.743X4>  encreafed by  add- 
[*nS  3  for  4X 8 

52.74)8397 
5274 

52.7)3123 

2637=5 27  X5>  encrtafed  by  adding 

« •  [*f or  5X4 

52)486 

473—52X9,  increajed  by  adding  5 


IO 

3  P  Required 


(       "4      ) 

Required  the  quotient  of  780.516-7-24.3,  in  inte 
gers  only. 

OPERATION. 
24\78o.5i6(32n:<7#0//V;tf  required. 

) 73=24X3,  increafed  by  adding  1/^3X3 

5  =  2X2,  increafed  by  adding  i  for  2X4 
o 


CHAP.       III. 

Of  REDUCTION  of  DECIMALS. 

PROBLEM     I. 

To  reduce  a  Vulgar  Fraction  to  its  equivalent  de 
cimal. 

RULE., 

ANNEX  cyphers  to  the  numerator,  and  divide  by 
the  denominator  till  nothing  remains,  and  the 
quotient  will  be  the  decimal  required. 

EXAMPLES. 

3 

Reduce  —  to  its  equivalent  decimal. 
20 

Thus,  30  \  3  .oo( .  i 5;=f  fo  decimal  required. 

)  20 


100 
IOO 


o  Reduce 


(      "5      ) 

i3 
Reduce  —  to  its  equivalent  decimal. 

20 
Thus,  20 \  i8.o(.9=/&<?  decimal  require/, 

I    ,Q~ 


)  180 
o 


6 

Reduce  —  to  its  equivalent  decimal. 

*5 

Thus,  I5\6.o(.4i=^  decimal  required. 
)  60 


^  8 
Required  the  equivalent  decimal  of  —  . 

9 

OPERATION. 


9^8.oo©oo(.8888  &V.  ad  infnitum. 


80 
72 

80 
72 

8 

HERE,  we  have  what  is  called  a  circulating  de 
cimal  for  the  quotient,  that  is,  a  continual  repetition 
of  the  fame  figure  without  any  poflibility  of  ever 
coming  to  an  end,  as  is  evident  from  the  example, 
Therefore  it  follows,  that  the  equivalent  decimal  of 
f  can  never  be  found  in  finite  ttfrms  ;  but  may  be 
obtained  to  any  degree  of  exa&nefs  you  pleafe. 

Note. 


(  Ilfi  ) 

Note.  When  a  vulgar  fraftion  is  annexed  to  a 
number  of  cents,  reduce  thefraftion  to  its  equiv. 
lent  decimal,  and  annex  it  to  the  tents,  and  t 
whole  will  become  a  decimal:  Thus,  tf^cen 

=•3775 

PROBLEM     II. 

?o  reduce  numbers  of  different  denominations  to  their 
equivalent  decimal. 

R  U  L  Jb?. 

REDUCE  the  given  numbers  to  their  equivalent 
vulgar  fraction,  by  problem  xvi  of  vulgar  fractions, 
then  proceed  as  in  the  laft  problem. 

EXAMPLES. 

REDUCE  3  qr.  2  na.  to  their  equivalent  decimal  of 
a  yard. 

Firft,  3  qr.  2  #0.r=~£of  a  yard  ; 

Then  i6\i4.ooo(.875=c:/fe  decimal  required. 


120 
112 


80 

80 


O 

4  b.  39 :'  10  "3  reduced  to  the  decimal  of  a 
187615  fcfr. 

8  S.  reduced  to  the  decimal  of  the  eclipticir.666 
.  ad  infinitum. 

o-iV  in.  reduced  to  the  decimal  of  a  foot^.9 


(      "7      ) 

£p.  reduced  to  the  decimal  of  an  acre=.o 333 

Ad  infinitum. 

PROBLEM     III. 

*To  find  the  value  of  a  decimal  in  known  parts  of  the 
integer. 

RULE. 

1.  MULTIPLY  the  given  decimal  with  the  parts 
in  the  next  inferiour  denomination,  and  point  off  as 
in  common   multiplication  of  decimals ;    and  the 
whole  numbers  will  be  the  value  of  the   given  de 
cimal  in  that  denomination. 

2.  MULTIPLY   the  remaining   decimal  with  the 
parts  in  the  next  mferiour  denomination,  and  point 
off  as  before,  and  fo  on,  thro  all  the  inferiour  deno 
mination,  if  need  be  i  and  you  will  have  the  value 
fought. 

EXAMPLES. 
Find  the  value  of  .875  of  a  yard. 

OPERATION. 
.875 
4 

3.500 

4 

••  ..   ,  qr.  na. 

a.ooo  tbmfore,  .875:1:3  2,  the  value  fought. 
Find  the  value  of  .426  of  a  pound  troy. 

OPER- 


.426 

12 


iyg       ) 
OPERATION. 


960 
480 

' 16  oz.  dwt.  gt\ 

•S-1&Q   fare/ore  .4*6  — $       2       5.76 

Find  the  value  of  .75  ofapound  fterling  of  Great- 
Britain. 

OPERATION. 

•75 

444 

300 
300 
300 

£.      cts.  do  I.  cts. 

333-00  therefore  .75-333-3  33- 

Find  the  value  of  .37752  of  a  pound  fterling  of 
Great-Britain. 

cts.         dol.       cts. 
Thus  .3775^X444-167.61888^:1  6761888. 

Note.  There  never  can  be  more  than  two  places  of 
cents,  and  where  there  are  other  figures  annexed, 
they  are  the  farts  of  another  cent :  thus,  in  the 
lafl  example,  the  6761 888m.  is  67  cents,  and 
.61888  of  mother.  A 


« 


A  SUPPLEMENT  to  PART    III, 

ONTAINING    THE     DOCTRINE     OF 

CIRCULATING  DECIMALS. 

CHAP.    I. 

DEFINITIONS  and  ILLUSTRATIONS. 

\  CIRCULATING  decimal  is  generated  or 
produced  from  a  vulgar  fraction,  vvhofe  nu 
merator  and  denominator  are  incommenfurable  to 
?ach  other  -,  and  therefore  if  the  numerator  with  cy 
phers  annexed,  be  divided  by  the  denominator,  there 
will  always  be  a  remainder,  or  the  quotient  will  run 
on  fempiternally  j  confequently  the  true  and  ade 
quate  decimal  of  every  iuch  vulgar  fraction,  muft 
confiftofan  infinite  number  of  decimal  places,  which 
is  therefore  not  aflignable  in  finite  terms,  and  confe- 
tjuently  the  true  and  complete  decimal  impofTible. 
NOTWITHSTANDING  the  equivalent  decimal  of 

very  vulgar  fraction  of  the  kind  above  defcribed, 
if  actually  completed,  would  then  confift  of  an  in 
finite  number  of  decimal  places  ;  yet  from  a  few  of 
the  firft,  we  obtain  fome  certain  law  by  which  the 
figures  ever  after  circulate  or  return  again  ;  and  it 
is  for  this  reafon  they  are  called  circulating  decimals  : 
the  circulating  figures  are  called  repetends,  of  which 
there  are  fowr  kinds,  viz.  fmgle,  compound,  mixed- 
fi^ngle,  and  mixed  compound. 

A  SINGLE  repetend  is  a  continual  repetition  of  tl^e 
fame  figure  :  Thus  .666  &c.  and  .2222  &V.  are  fm 
gle  repetends,  which  are  rxprcfTed  by  writing  the  re- 

pelting 


56  &c, 


peating  figure  with  a  point  over  it :  thus,  for  .666 

«  • 

write   .6  for  .2222  &V.  we  write  .2;  andfo  on  foi 
others. 

A  COMPOUND  repetend  is  when  the  fame  figures 
circulate  or  return  alternately  :  thus  .9595  '&c.  and 
.321321  &c.  are  compound  repetends,  which  areex- 
prefTed  by  writing  the  combination  of  figures  that 
circulate  or  return  together,  with  a  point  over  the 
firft  and  lad  figure  :  thus,  inftead  of  -959J  &c.  we 

write  .95  for  .321321  &c.  we  write  .321  ;  and  fo 
on  for  others. 

A  MIXED  fingle  repetend  is  when  one  or  more  fi 
gures  occur  before  the  repeating  ones  :  .thus  .172444 
&c,  and  .1942777  &V.  are  mixed  fingle  repetends. 

A  MIXED  compound  repetend  is  when  fevcral  fi 
gures  (land  before  thofe  that  circulate  alternately: 
thus  .1724747  &c.  and  .41972972  &c.  are  mixed 
compound  repetends, 

THOSE  combinations  of  figures,  which  circulate 
or  return  together,  are  called  circu!ates,  of  which 
there  are  three  kinds,  viz.  fimilar,  diflimilar,  fimijar 
and  conterminous. 

SIMILAR  circulates  are  thofe  that  confift  of  the 
fame  number  of  repeating  figures,  beginning  either 

before  or  after  the  decimal  point:  thus  42.7  and  9.19 
are  fimilar  circulates. 

DISSIMILAR  circulates  are  thofe  that  confift  of  an 
unequal  number  of  repeating  figures,  beginning  at 

different  places  :  thus   1.77  and  217.4  are  diflimilar 
circulates* 

SIMILAR 


SIMILAR  and  conterminous  circulates,  t  arc  thole 
which  confift  of  an  equal  number  of  repeating  fi 
gures,  beginning  and  ending  together  :  thus,  27.47 
and  4.73  are  fimilar  and  conterminous  circulates. 


CHAP.      II. 

Of    REDUCTION  of  CIRCULATING    DECI 
MALS. 

PROBLEM     I. 

To  reduce  a  fingle  repetend  to  its  equivalent  Vulgar 
Fraffion. 

RULE.      , 

UNDER  the  given  repetend,  with  as  many  cy 
phers  annexed  to  it,  as  there  are  places  of  whole 
(numbers,  write  as  many  9^  as  there  are  places  of 
figures  in  the  repetend  j  and  you  will  have  the  Vul 
gar  Fraction  required. 

THE  reafon  of  this  rule  will  appear  obvious,  when 

weconfider,  that  .9—1 ;  for  fz=.  in  &V.— .1  j  con- 

fequently  ,ix9~iX9  ;  that  is,  .9— f-:± i  ;  whence 
it  follows,  that  each  figure  of  the  repetend  is  equal  to 

that  figure  divided  by  9  :  thus  .3— £=4  ,5=:  j.,&fr. 
EXAMPLES, 

Required  the  leaftVulgarFra&ion  equivalent  to  .72 

Thus, 


Thus,  ^-zz^^f raft  ion  required. 

•  •      21300        ^        •      64^2^000 

21.3^1  — g 643.25=-2£-2 1.7421= 

999  .  99999 

174.210     -  1270002000 

-  -.   127.0002= — ; . 

99999  9999999 

PROBLEM     II. 

fo  reduce  a  mixed  compound  repetend  to  its  equivalent 
Vulgar  Fraction. 

RULE. 

WRITE  down  as  many  9'^  as  there  are  places  of 
figures  in  the  repetend,  to  which  annex  as  many  cy 
phers  as  are  equal  to  the  number  of  occurring  places 
of  figures  in  the  finite  part,  (i.  e.  the  figures  occurring 
before  the  alternate  circulates)  for  a  denominator  ; 
then  multiply  the  9'^  in  the  denominator,  with  the 
finite  part,  to  which  product,  add  the  infinite  or  cir 
culating  part  for  a  numerator  j  and  you  will  have; 
the  fraction  required. 

Note.  When  the  circulate  begins  any  where  in  the 
integral  fart,  omit  the  cyphers  in  the  denominator, 
and  annex  as  many  to  the  numerator  as  there  are 
places  of  whole  numbers  included  in  the  circulate. 

TnEreafon  of  this  rule  will  appear  plain  from  the 
following.  Suppofe  the  decimal  whofe  equivalent 

VulgarFraftion  is  required,  to  be  .53:  Conceive  it 
to  be  divided  into  finite  and  infinite  parts  -,  that  is, 
conceive  it  to  be  made  of  the  finite  part  .5  and  the  in 
finite  or  circulating  part  .03  j  then  .53  =  .54-.O3j  but 
.3:r~i  confequently  .ojZZ-rV  of  -J-=7V»  wherefore 

-53 


53=  =+ 


(      i*3      ) 

«« 

-£i^>  which  is  the 
90 

fame  as  the  rule. 

EXAMPLES. 

Required  the  Vulgar  Fraction  equivalent  to  .4739 
Firft,  9990— denominator. 

Then  999X4-:=:3996—prodi{ff  of  the  cfs  in  the  de 
nominator  and  finite  part -,  and  3996+739—47351= 
numerator. 

Wherefore  -f-J-14  is  the  fraction  required. 

Required  the  equivalent  Vulgar  Fradtion  of  5.27  : 

Thus,  52X9+7 -r  900=468+7-*- 900=^54    tbt 
fraction  required. 

Required  the  equivalent  Vulgar  Fraction  of  42.3  : 

Thus,   990X4+230-7-99—  4£f.°  the  fraftion  re 
quired. 

Required  the  equivalent  Vulgar  Fraction  of  321.7  : 

Thus, "999x3  +  217-7-999  =  3214-7-999  j    then 
"*  ^frdSlon  required* 


14-00 


PROBLEM     III. 

To  determine  whether  the  decimal  equivalent  to  any 
Vulgar  Fraction  be  finite,  or  infinite  -,  and  if  infinite  y  to 
find  the  number  of  places  of  figures  that  ccnftitute  the 
circulate. 

RULE. 

1.  REDUCE  the  given  fraction  to  its  lead  terms. 

2.  DIVIDE  the  denominator  of  the  refulting  frac 

tion 


(         124        ) 

tion  by  i,  5  or  10,  as  often  as  you  can  without  a  re 
mainder,   making  the  refult  a  divifor,  and  999  &V. 
a  dividend,  divide  till  nothing  remains,  then  will  the 
circulate  confift  of  as  many  places  of  figures  as  you  j 
ufed  places  of  ffs. 

Note,  i .  The  circulate  will  begin,  after  as  many 
places  of  figures  as  you  made  divifions  of  the  de 
nominator. 

2t  In  dividing  the  denominator  as  above,  if  the  quo 
tient  become  equal  to  unity,  then  the  decimal  is 
finite,  confining  of  as  many  places  of  figures  as  you 
made  divifions  of  the  denominator. 

THE  principles  on  which  this  rule  is  inveftigated, 
may  be  fhewn  in  the  following  manner. 

Firft,  let  it  be  premifed,  that  if  unity  with  cy 
phers  annexed,  be  divided  by  any  prime  number, 
except  2,  or  j,  the  figures  in  the  quotient  will  begin 
to  repeat  when  the  remainder  becomes  unity ;  con- 
fequently  999  &c.  divided  by  any  prime  lumber, 
except  2,  or  5,  will  leave  no  remainder. 

Now  if  the  places  of  figures  in  the  circulate  are 
any  number,  when  the  dividend  is  unity,  they  will 
remain  the  fame,  let  the  dividend  be  any  other  num 
ber  whatever  ;  for  it  is  plain,  that  if  the  decimal  be 
multiplied  with  any  number,  every  circulate  will  be 
equally  multiplied,  and  what  one  is  increafed  will  be 
carried  to  another,  and  fo  on  through  the  whole  ; 
confequently,  the  places  of  figures  will  remain  the 
fame  :  But  to  multiply  the  decimal  or  quotient  with 
any  number,  is  the  fame  thing,  as  to  divide  the  divi- 
ibr  by  the  fame  number  before  divifion  is  made; 
whence^  &c. 

EXAMPLES.. 


EXAMPLES, 


Required  to  know,  whether  the  equivalent  deci 
mal,  of-'-rr  is  infinite  or  finite,  and  if  infinite,  how 
many  places  of  figures  there  will  be  in  the  circulate. 

Firft,  -i-f4  reduced  to  its  lead  terms ~£  ;  then 
999999-7-7=142857,  and  therefore  the  decimal 
is  infinite,  whofe  circulate  confifts  of  6  places  of  fi 
gures,  beginning  at  the  tenth's  place. 

Required  to  know  whether  the  equivalent  dicimal 
of  -i-V-sSr  is  infinite,  or  finite  ;  and  if  infinite,  how 
many  places  of  figures  there  will  be  in  the  circulate. 

Firft,  -m^=(by  reducing  to  its  lead  terms)  Ty  ; 
then,  i6-r-2n  8,  8 -r-2 —4,  4—2—2,  and  2  —  2=1  : 
Consequently  the  decimal  is  finite,  confifting  of  4 
places  of  figures. 

-Required  to  know  whether  the  equivalent  decimal 
of  l|t  is  infinite,  or  finite  ;  and  if  infinite,  to  know 
how  many  places  of  figures  there  will  be  in  the  cir 
culate. 

Firft,  |44 —  (by  reducing  to  its  leaft  terms)  44  ; 
then  70-7-10=7,  and  999999-r-7:::::::i42857  :  Con- 
fequently  the  dicimal  is  infinite,  and  the  circulate 
confifts  of  6  places  of  figures,  beginning  at  the  hun 
dredth^  place. 

PROBLEM      IV. 

To  make  diffimilar  circulates  >  fimilar  and  conterminous. 
RULE. 

i.  Find  the  leaft  poffible  common  multiple  of  the 
feveral  numbers  exprefRng  the  number  of  places  of 
figuies  in  the  given  circulates, 

2. 


2.  Change  the  given  circulates  into  others.confift- 
ing  each  of  as  many  places  of  figures  as  the  leaft 
common  multiple  found  as  above^  and  the  work  will 
be  done. 

EXAMPLES. 

•    .     »    •      •  <•          •  * 

Make  .727,  .179,  .12  and  .19  fimilar  and  con 
terminous. 

Firft  the  leaft  common  multiple  of  3,  3,  2  and  2, 
IE  6. 

Diflimilar.  Similar  and  conterminous, 

f -7  27  ~-7  277  27 
Then,  j-?79=.i79i79 

j.12    =.121212 
1.19   =.191919 

Make  24.3,  .4762,"  32,  ,6  and  .576  fimilar   and 
conterminous. 

Diffimilar.    Similar  and  conterminous, 

^24.3=24.333333333333 
Thus  \  '^7 62=. 47 6 247 6 '2-47 6*2 
9  j  32.6=32.666666666666 
1.576^.576576576576" 


C  PI  A  P. 


CHAP.      III. 

ADDITION,  SUBTRACTION,  MUL 
TIPLICATION  AND  DIVISION  OF 
CIRCULATING  DECIMALS. 

SECT.       I. 

Of     ADDITION    of    CIRCULATING 
DECIMALS. 

R  ULE. 

MAKE  the  given  circulates  fimilar  and  conter 
minous,  by  problem  iv,  of  the  lail  chapter; 
then  add  them  together  as  in  common  Addition,  and 
becaufe  each  figure  of  the  circulate  is  equal  to  that 
figure  divided  by  9,  you  mufl  divide  the  fum  of  the 
circulates,  by  as  many  places  of  $'s  as  there  are  pla 
ces  of  figures-  in  the  circulate,  and  writing  the  re 
mainder  (if  any)  directly  beneath  the-figures  of  the 
circulate,  carry  the  above  quotient  to  the  next  place  ; 
then  proceed  as  in  common  decimals,  and  you  will 
have  the  fum  required. 

Note.  When  the  remainder  confifts  of  a  hjs  number 
of  places  than  the  circulate,  you  miift  fitpfly  !bt 
defeffi  ly  prefixing  cyphers. 


EXAMPLF3, 


(          128         ) 

EXAMPLES. 

Required  the  fum  of  3.3+4.271+3.725  : 

Similar  and  conterminous. 


!3-3  .=3-333 
4.271=4.271 
3-725=3.725 


ii.330=r///ff*  required. 

Required  the  fum  of  24,327425  +  37.274+27.35+ 

34-27  : 

Diffimilar.     Similar  and  conterminous, 

^24.327425  —  24.327425425 

Thus,  I  37-274  =  37-274444444 
127.35*  =27.353535353 
134.27  =34.277777777 

•  * 

1 23. 233 1 8  300 1  ~y#w  req, 

SECT.       II. 

Of  SUBTRACTION  of  CIRCULATING  DECI 
MALS. 

RULE. 

PREPARE  the  given  numbers,  as  in  Addition,  and 
then  fub  tract  them  as  in  common  SubtradHon,  only 
with  this  difference,  viz.  when  the  circulate  to  be  fub- 
tra&ed,  is  greater  than  the  one  from  which  Subtrac 
tion  is  to  be  made,  you  muft  make  the  right-hand 

figure 


figure  of  the  difference  lefs  by  unity,  than  as  found  by 
common  Subtraction.  The  reafon  of  this  rule  will 
appear  plain  from  the  following. 

SUPPOSE  1.81  were  to  be  taken  from  2.72;  the 
difference  by  common  Subtraction  would  be  .91  -t  but 
2.72=  Vs?  and  i.8i~VV%  then  2.72^1.81:=: 
—  VTO==£?==  .90;  whence, 


EXAMPLES. 

Required  the  difference  between  6.4729  and  3-49': 
Diflimilar.     Similar  and  conterminous. 
29-6.4729729 


^  |6.47  29:1:6., 
''  U'49     =3- 


Thus 

,4949494 


•  » 

2  ,  97  8  0234^  difference  required. 

Required  the^  difference  between  4.  37  5  2  and  1.1210  : 
Diffimilar.     Similar  and  conterminous* 


Th 

US>        I.iaiO=:I.I2lOl2IO 


3,25424041=  difference  required. 

SECT.      III. 

Of  MULTIPLICATION  of  CIRCULATING  DE~ 
CIMALS. 

RULE. 

INSTEAD  of  the  given  circulates,  write  their  equi 
valent  Vulgar  Fraftions,  and  find  their  product  as 

R  ufual  ; 


ufual ;  then  this  produft  thrown  into  a  decimal,  will 
give  the  product  required. 

EXAMPLES. 

Required  the  product  of  3.2X-7 

Firft,  .32— -Jl- and  .7— £>  wherefore   .jix.?  — 
which    thrown    into    a    decimal   is, 

ra/«#  required. 
Required  the  product  of  1.8x2.7  : 

Thus,    i.^2.7-V7XV5=\V-5'S 
the  pro  duff  required. 

Required   the   product  of. 20X^36  : 

Thus,  .2oX.36~r^X^4=^4 
produft  required. 

SECT.       IV. 

Of  DIVISION  of  CIRCULATING  DE 
CIMALS. 

R  ULE. 

CHANGE  the  given  decimals  into  their  equivalent 
Vulgar  Fractions,  and  find  their  quotient  as  ulual  ; 
then  this  quotient  thrown  into  a  decimal,  will  give 
the  quotient  required. 

EXAMPLES. 

•  • 

Required  the  quotient  of  .26  divided  by  .3  : 

3-~  ' 

Wherefore, 


Wherefore,  .26^.3=14-H-=T74=.8  tie  quotient 
required. 

Required  the  quotient  of  .9-7-.! 08  : 


Thus,    .9^08:^-^=^  AV=VT  =9.25 
^quotient  required. 

Required  the  quotient  of  2. 9-4-, 27* : 

Thus,    2.9~.27:=y^^:rY'==  n  the  quotient 
required. 


13* 


A  SUPPLEMENT  to  PART    I, 

CONTAINING  THE  DOCTRINE  AND 
APPLICATION  OF  RATIOS,  OR 
PROPORTION,  EXTRACTION  OF 
ROOTS,  C5>, 

CHAP,     L 

Of  PROPORTION  or   ANALOGS 

PROPORTION  is  a  degree  of  likenefs  which 
quantities  bear  to  each  other,  by  a  fimilitude 
of  ratios. 

RATIO  is  the  mutual  refpect  of  two  quantities  of 
the  fame  kind  ;  but  they  form  no  Analogy,  becaufe 
there  can  be  no  fimilitude  of  ratios  between  two 
quantities,  and  therefore  Analogy  confifts  of  three 
quantities  at  leaft,  whereof  the  iecond  fupplies  the 
placeof  two  :  Thus  the  refpect  of  2  to  63  being  com 
pared  with  1 8,  it  will  be,  2'.6::6:i8. 

SECT,      I. 
Of  CONTINUED    PROPORTION 


O  R 
ARITHMETICAL   PROGRESSION, 

WHEN  quantities  increafe  or  decreafe  by  an  equal 
difference,  thofe  quantities  are  in  Arithmetical  Pro 
portion  continued  :*  Thus,  the  number  i,  2,  3,  &c, 
are  a  feries  of  quantities  in  Arithmetical  Proportion 

continued., 


(       133-     X 

continued,  increafing  by  unity,  or  i,  which  is  called 
the  common  difference  of  the  feries. 

ALSO,  the  numbers  2,  4,  6,  8,  are  numbers  in 
Arithmetical  Progreflion,  whofe  common  difference 
is  2  ;  but  the  numbers  9,  7,  5,  3,  i,  are  a  feries  of 
quantities  in  Arithmetical  ProgrefTion,  decreafing  by 
the  common  difference,  2. 

LEMMA    I. 

If  three  numbers  are  in  Arithmetical  Progreffion,  the 
fum  of  the  two  extreme  numbers  will  be  double  the  mean 

or  middle  number. 

THUS,  let  i,  3,  5,  be  the  numbers  in  progreffion ; 

Then,  1+5,  the  fum  of  the  two  extremes  ~  3-^3 
the  double  of  the  mean.  Again,  in  the  numbers 
14,  ic,  6,  the  fum  of  the  two  extremes  are  i4-f  6~ 
20,  and  the  double  of  the  mean  10+10—20  ;  and 
the  like  will  hold  in  any  other  numbers. 

LEMMA     II. 

If  four  numbers  are  in  Arithmetical  ProgreJ/ion,  the 
Jam  of  the  two  extremes  will  be  equal  to  the  fum  of  tbc 
two  means. 

LET  the  number  be  4,7,  10, 13 ;  then4~f-i3~i7j, 
the  fum  of  the  two  extremes,  and  7~|-io:i:i7,  the 
fum  of  the  two  means  :  Again,  in  the  numbers  16, 
13,  10,7  i  16+7  —  13+10. 

AND  iince  in  four  numbers  as  above,  the  fum  of 
the  two  extremes,  is  equal  to  the  fum  of  the  two 
means,  we  have  no  reafon  to  doubt  of  the  like,  let 
the  terms  be  any  number  whatever  :  Whence  it  fol 
lows,  that  in  any  Arithmetical  feries,  of  any  afllgna- 
ble  number  of  terms  whatever,  the  fum  of  any  two 
terms  equidiftant;  from  the  mean,  will  be  equal  to 

the 


the  fum  of  any  other  two  terms,  equidiftant  from  the 
mean  j  as  in  thefe,  2,  4,  6,  8,  10,  12,  14,  16,  18,  20 ; 
where  2-f-2O=4-f  18^:6^16=: 8+14=10+1 2  : 
Therefore,  &V. 

LEMMA    III. 

/#  any  Jeries  of  numbers  in  Arithmetical  Progreffion^ 
thejeveral  terms  are  formed  or  made  up  by  the  addi 
tion  of  the  common  difference  to  the  frft  term,  fo  often 
repeated,  as  there  are  number  of  terms  to  the  feveral 
places,  except  the  firft. 

LET  the  feries  be,  i,  4,  7, 10,  13,  16,  19,  22,  fc?r* 
wherein  the  common  difference  is  3. 

Now  1+3—4  the  Jecond  term,  i+jH-3~7>  the 
third  term  \  i-r-3+3*f-3  — 10,  the  fourth  term  \ 

J*3-Klf  3-f3=i3»  **'  fifth  term  i  and  1+3X7 
r:22,  the  %th  term,  &c.  Confequently  the  differ 
ence  of  the  two  extremes,  is  equal  to  the  common 
difference  multiplied  with  the  number  of  terms  lefs 
i  :  Thus  in  the  above  feries,  the  common  difference 

is  3,  and  number  of  terms  8  ;  therefore  8— IX3—7 
X  3—2 1  rz difference  of  the  two  extremes. 

PROBLEM     I. 

¥0  find  thejum  of  a  Jeries  of  numbers  in  Arithmeti 
cal  Progrejfion. 

THERE  are  feveral  ways  of  deducing  a  rule  for  the 
folution  of  this  problem,  but  perhaps  none  more  fim- 
ple  and  natural  than  the  following. 

LET  the  feries  whofe  fum  is  required,  be  2+4+6 
-fS-fio-f  12. 

Or, 


Or, 


22222 

+  +  +  + 

2222 

+  +  + 

222 

4-   + 

2       2 


which  is  the  fame  as  the  former,  though  differently 
exprefTed  :  Now  under  the  given  feries  place  die 
fame  inverted  and  add  up  the  whole. 


Thus, 


~    HI 

10 

Co 

t. 

o^ 

? 

1 

i 

P 

i 

1 

i 

2    -f 

2    -f 

'    ^    + 

2  -f 

2     - 

L    2 

2 

4- 

+ 

+ 

+ 

4- 

4- 

2 

2 

2 

2 

2 

2 

2 

4- 

4- 

4- 

4- 

"T" 

4" 

2 

2 

2 

2 

2 

2 

2 

4 

+ 

4- 

~f- 

f 

+ 

2 

2 

2 

2 

2 

2 

2 

-f- 

•f 

~f- 

+ 

+ 

"f 

2 

2 

2 

2 

2 

o 

2 

4- 

f 

-f- 

•f 

+ 

-f- 

2 

12 

2 

2 

2 

- 

2 

14+  14+  14  +14  -f  14  HhJ4=^*- 

By  this  means  the  terms  of  the  feries  are  reduced 
to  an  equality,  to  wit,  equal  to  the  fum  of  the  firll 
and  laft  term  ;  but  the  fum  above  found,  is  evident 
ly  double  the  fum  of  the  propofed  feries  :  When  a; 

it 


C 


it  folloxvs,  that  the  fum  of  an  Arithmetical  feries,  is 
equal  to  half  the  product  of  the  firft  and  laft  term, 
with  the  number  of  terms;  wherefore  if  the  firft  term, 
jail  term,  and  number  of  terms  of  an  Arithmetical 
ProgrefTion  be  given,  the  fum  of  the  feries  may  be 
found  by  the  following 

RULE, 

MULTIPLY  the  fum  of  the  firft  and  laft  terms,  or 
two  extremes,  with  the  number  of  terms,  and  half  of 
that  product  will  be  the  fum  required. 

EXAMPLES. 

Let  the  firft  term  of  a  feries  of  numbers  in  Arith 
metical  Progrcilion,:~i,  Lift  termizjy,  and  num 
ber  of  terms  19  -3  required  the  fum  of  the  feries. 

OPERATION. 

Firft,  i  ^37=3  8  =:/«/»  of  the  frfl  and  laft  terms  : 
Then  3&X  i9-~2~  722-t-2~j6i  the  Jum  required. 

A  MAN  bought  20  yards  of  broad-cloth  -,  for  the 
firft  yard  he  gave  2  doL  and  for  the  laft  80  dol. 
what  did  the  whole  coft  ? 

The  fum  of  the  two  extremes,    is  2-/-8o,  then 

2  -f-S  ox  20  -7-2=1:820  dol.  the  anjwer. 

A  MAN  travelled  12  days,  the  firft  day  4  miles,  and 
the  laft  day  40  miles  j  what  was  the  diftance  travelled 
in  the  1  2  days  ?  Anfwer.  264  miles. 

PROBLEM     IL 

To  find  the  common  difference  of  an  Arithmetical  Je- 
riesy  when  the  two  extremes  and  number  of  terms  are 

given.  .  A 


A  RULE  for  the  folution  of  this  problem,  is  eafily 
deduced  from  the  inference  to  Lemma  in  5  for  fince 
the  difference  of  the  two  extremes,  is  equal  to  the 
common  difference  multiplied  with  the  number  of 
terms  lefs  i,  it  follows,  that  if  that  difference,  be 
divided  by  the  number  of  terms  le&  i,  the  quotient 
muft  be  the  common  difference  of  the  feries  ;  whence 
the  following  rule  is  evident. 

RULE. 

DIVIDE  the  difference  of  the  two  extremes  by  the 
number  of  terms  lefs  i,  and  the  quotient  will  be  the 
common  difference  required. 

EXAMPLES. 

In  an  Arithmetical  feries,  there  is  given  the  fir  ft 
term—  3,  lad  term—  60,  and  number  of  terms  i^: 
Required  the  common  difference. 

OPERATION. 

The  difference  of  the  two  extremes,   is  60—  3; 
^_I--^  common  difference 


20—1 

required. 

Four  men  differing  in  their  ages  by  an  equal  in 
terval  :  The  age  of  the  firft,  is  1  9  years,  *nd  the 
fourth  40  :  What  are  their  feveral  ages  ? 

OPERATION. 

Firft,  find  the   common   difference  of  their  ages  : 
Thus,    40  —  1  9^-4^7=2;  i  -4-3=7  ?"*rs  ;  therefore 

S  i 


<      138      ) 

—  z6  years,  the  age  ofthefecond,  and  26-1-7=33 
years,  the  age  >cf  ^the  third-,  laftly,  33-j~7rr4O  years, 
the   age  of  tbe  fourth,  as  given  above. 

A  man  owes  a  certain  debt,  to  be  difcharged  at 
8  feveral  payments  ;  all  of  which  are  to  be  made  in 
Arithmetical  Progreffion,  the  firft  payment  to  be  4 
dol.  and  the  Jail  32  dol.  Query,  the  whole  debt  and 
each  payment. 

OPERATION. 

wbole  debt,  and  32—4-7- 


1  1=4  dol.  the  common  difference  -,  wherefore  4+4 
^=8  dol.  the  Jecond  payment,  and  8-j-4~i2  dol.  the 
third  payment  •>  alfo,  12+4:1116  dol.  ;  for  the  fourth-, 
moreover  1  6  +4:1:20  dol.  for  the  $th,  in  like  man 
ner  20  -f  4—24  dol.for  the  6th,  and  24-j-4=28  dol.  for 
the  jtb  -,  laftly  284-4—32  dol.  for  the  laft  payment 
as  before. 

PROBLEM     III. 

To  find  the  number  of  terms  of  an  Arithmetical  Jeries, 
when  thefirjl  term,  laft  term  and  common  difference  are 
given. 

FROM  the  laft  rule,  it  is  eafy  to  conceive  haw  a 
rule  for  the  folution  of  this  problem  may  be  obtain 
ed  i  for  fince  the  difference  of  the  two  extremes, 
divided  by  the  number  of  terms  lefs  i,  gives  the 
common  difference  ;  it  follcfcsj  that  the  difference  of 

.V 

the  two  extremes,  divided  by^the  common  difference, 
muft  quote  the  number  of  terms  lefs  r  . 
Whence  is  deduced  the  following 

RULE. 


RULE. 

DIVIDE  the  difference  of  the  two  extremes,  by  the 
common  difference,  the  quotient  increafed  by  unity 
or  i,  will  be  the  number  of  terms. 

EXAMPLES. 

Given  the  firft  term  of  an  Arithmetical  ferieszrs, 
laft  termm67,  and  common  difference  3,  to  find  the 
number  of  terms. 

OPERATION. 


±+I=Ji+  1=55+1-56  the  nmhr  of 

3     .     .      3 
terms  required. 

A  man  bought  a  quantity  of  broad-cloth  ;  for  the 
firft  yard  he  gave  6  dol.  for  the  fecond,  10  dol.  and 
fo  on,  in  Arithmetical  Progreffion,  to  the  laft  yard, 
for  which  he  gave  246  dol.  j  what  was  the  quantity 
of  cloth  bought  ? 

_  OPERATION. 

046—6       =240+  I=6i,  the  numlerof  yards  bought. 

4  ~4 

A  man  travels  from  Bofton,  to  a  certain  place,  in 
the  following  manner,  viz.  the  firft  day  10  miles  ;  the 
fecond  day  15  miles,  and  fo  on,  till  a  day's  journey 
is  cc  miles  :  In  how  many  days  will  he  perform  the 
whole  journey  ;  alfo,  how  many  miles  is  the  place 
he  goes  to,  diftant  from  Bofton  ? 

Anfwer.  Pie  will  perform  the  whole  in  eleven  days, 
The  place  diftant  from  Bofton,  330  miles. 

SECT. 


SECT.      II. 

Of    CONTINUED     PROPORTION 
GEOMETRICAL, 

Or 
GEOMETRICAL    PROGRESSION, 

GEOMETRICAL  Progreifion  continued,  differs  frorr 
Arithmetical  Progreflion  in    this  -,  in  Arithmetica 
Progreifion,  each  following  term  of  the  feries  is  form 
ed  or  made   up  by  the  Addition  or  Subtraction  o 
the  common  difference,  (as  we  have  before  fhewft) 
Whereas  in  Geometrical  Progreffion,  each  fucceffive 
term  of  the  feries,  is  produced  by  the  Multiplication 
or  Divifion  of  the  preceeding  term,  with  a  common 
multiplier  or  divifor  :  Or  in  other  words,  Arithme 
tical  Progreflion,  is  the  effect  of  a  conftant  Addition 
or  Subtraction  -,  but  Geometrical  Progreffion,    of  a 
conftant  Multiplication  or  Divifion. 

THUS,  2,  4,  8,  1  6,  32,  64,  128,  &V.  are  a  feries 
of  numbers  in  Geometrical  Proportion  continued  ; 
whofe  refpective  terms  are  compofed  by  the  Multi 
plication  of  the  Ratio  or  common  multiplier,  (2): 
thus,  2X2:1=4,  the  Jecond  term,  4X2^18,  the  third 
term  ;  8X2:1:16,  the  fourth  term)  and  fo  on. 

ALSO,  1  6,  8,  4,  2,  are  a  feries  of  numbers  in  Geo 
metrical  Proportion,  continually  decreafmg  by  the 
divifion  of  the  Ratio,  or  common  divifor,  (2)  :  Thus, 
-^i^S,  the  Jecond  term>  4^:4,  the  third  term,  ^2,  the 
fourth  term,  and  T—1*  ifajtjftt  term. 


LEMMA     I. 

Jf  three  numbers  are  in  Geometrical  Progtefficn,  the 
froduSt  cf  the  two  extremes,  will  le  equal  to  the  pro- 
duff  of  the  mean  with  iff  elf. 


(    HI    ) 

LET  the  numbers  be  2,  8,  32 ;  where  2X32—64,  and 
8X8=64  5  confequently  2X32=8X8. 

LEMMA     II. 

In  any  Geometrical  Proportion  fonftfting  of  four 
terms,  the  produfl  of  the  two  extremes,  is  equal  to  the 
pro  duff  of  the  two  means. 

IF  the  numbers  are,  2,  8,  32,  128,  it  will  be 
2x128:118x32  j  therefore  2:  8  1:32  1128. 

CONSEQUENTLY,  if  the  product  of  any  two  num 
bers,  be  equal  to  the  produd:  of  any  other  two  num 
bers,  thofe  four  numbers  are  proportional. 

HENCE  it  may  be  eafily  underftpod,  that  if  any 
number  of  terms  are  in  -ff  the  product  of  the  two  ex 
tremes,  will  be  equal  to  the  product  of  any  other 
two  terms,  equidiftant  from  thofe  extremes. 

LET  the  feries  be  3,  6,  12,  24,  48,  96  ;  where 
3X96=6X48-12X24. 

WHEN  numbers  are  compared  together,  in  order 
to  difcover  their  relation  to  each  other,  the  number 
compared  is  writen  firft,  and  called  the  antecedent, 
and  the  number  by  which  you  compare  the  other, 
being  written  next,  is  called  the  confequent  :  Thus 
if  you  would  compare  2  with  4,  the  numbers  muft 
be  wrote  thus,  2,  4  ;  where  2  is  the  antecedent,  and 
4  the  confequent  :  Again  in  theie,  3  :  6  ::  6:  12  ; 
where  3  is  antecedent,  and  6  its  confequent  ;  alfo, 
6  the  middle  term,  is  an  antecedent  to  12,  its  confe 
quent.  Therefore  in  every  feries  of  numbers  in 
Geometrical  Proportion  continued,  all  the  terms  ex 
cept  the  laft,  are  antecedents,  and  all  except  the  firft 
are  confequtnrs. 

THUS  in  (fee  feries  3,  9,  27,  81,  243,  729,  the 
numbers  3,  9,  27,  81,  243,  arc  all  antecedents,  and 


•9,  27,  8 1,  243,  729, 


i  all  confequents  ;  therefore 
8 1 ::  81:243::  243:729. 
THE  Ratio  is  had  by  dividing  any  confequent  by 


3  19  1:9  127::  27 


its  antecedent^ 


LEMMA      III. 


If  any  numbers  are  proportional)  it  will  be,  as  any 
cne  cf  the  antecedents,  is  to  its  confequent  jfo  is  the  fum 
of  all  the  antecedents,  to  to  the  fum  of  all  the  confe- 
qusnts,  (!yid.  Euclid's  fifth  book,  Proportion  12.) 

LET  the  numbers  be  thefe,  4,  8,  16,  32,  64,  then 
4-32  :  84-164-32  +  64,  that  is, 
for  4Xi20iz8X6o  j  therefore, 


8  ::  44-84- ic 
8  ::  60  :  1 20  j 


PROBLEM      I. 


To  find  tl}3jum  of  any  Geometrical  Jeries  increofing. 

SUPPOSE  the  fum  of  the  following  feries,  i,  4,  16, 
64,  256,  is  required  :  Multiply  this  feries  with  the 
Ratio,  which  is  4,  and  the'  product  will  be  a  new  fe 
ries,  4,  1  6,  64,  256,  1024  :  Now  it  is  plain,  that 
the  fum  of  the  produced  feries,  is  as  many  times  the 
fujQDL  of  the  former,  as  the  Ratio  hath  units  ;  or  the 
produced  feries,  is  to  the  propofed,  as  the  Ratio  to 
unity,  or  i  :  Subtract  the  firft  feries  from  the  fccond, 

4>  l6>64>  256>  I024 
1,4,  1  6,  64,  256. 


Thus 


• 


i,   '  •    4-1024,        or,    1024  —  i, 

which  is  evidently  equal  to  the  fum  of  the  firfc  ftries 
multiplied  with  the  Ratio,  lefs  i,  by  what  has  been 
faid  ;  confequently  the  fame  divided  by  the  Ratio, 
lefs  i,  muft  give  die  fum  of  the  propofed  feries;  that 

is; 


THEREFORE,  when  the  firft  term,  lad  term,  and 
Ratio  of  a  Geometrical  feries  are  given,  we  may  find 
the  fum  of  all  the  terms  by  the  following 

RULE. 

MULTIPLY  the  lad  term  with  the  Ratio,  from 
which  produftj  fubtradfc  the  firft  term,  divide  the  re 
mainder  by  the  Ratio  lefs  i,  and  the  quotient  refuh- 
ingwill  be  the  fum  cf  the  fen cs. 

MR.  WARD,  in  his  introdu&ion  to  the  Mathe 
matics,  page  78,  has  given  an  analytical  invefiiga- 
tion  of  a  rule  for  rinding  the  fum  of  any  feries  infr 
increafing  ;  which  is  afcer  the  manner  following. 

LET  a  Geometrical  feries  be  given^  fuppofe  the 
following,  2,  4,  8,  16,  32,  64. 

Put  x~^Jum  of  the  feries  : 

Then,  x* — 64.— fum  of  all  tbe  antecedents  : 

And  x*—i^Jum  of  all  the  confequents  : 

Therefore,  2:4::  #—64  :x — 2  ;  per  Lemma 
in.  .  

Cohfeqticntly,  x — 2X2i±tf — 64X4* 

That  is,  2* — 4n4.v — 256  : 

Then,  4^- — 2^^256^ — 4  : 

Therefore,  (by  divifion)  ix — #1^1128 — 2  : 

Whence,  Arni28 — 2-r-2 — i,  wbich  affords  thefamf 
rule  as  that  above. 

Or  finding  tbe  value  ofx  in  tbe  equation  4*— 2#:n 

256 — 4,    to  wit)  ,VH256— 4~-4 — i    which  admits  of 
&e  following 

R  U  L  E, 


(       144-      ) 


RULE. 

FROM  the  produ6b  of  the  fecond  and  laft  terms, 
fubtracfc  the  fquare  of  the  firft,  divide  the  remainder 
by  the  fecond  term  lefs  the  firft  -,  and  the  quotient 
will  be  the  fum  of  the  feries. 

EXAMPLES. 

In  a  Geometrical  feries,  there  is  given,  the  firft 
termzz3,  laft  term=r243,  and  Ratio  3  j  to  find  the 
fum  of  the  feries,  per  Rule  firft. 

OPERATION. 

. 

Firft,  243  X  3  —  7  2  9 ^'prcduft  of  the  laft  term  with 
the  Ratio-,  then  729 — 3-7-3—1^726-7-2  =  363  the 
fum  required. 

A  man  bought  a  quantity  of  cloth  ;  for  the  firft 
yard  he  gave  2  dol.  for  the  fecond  4 ;  and  fo  on,  in 
continued  proportion  Geometrical  to  the  laft  yard, 
for  which  he  gave  256  dol.  what  did  the  whole  coft  ? 

Here,  is  given  the  firft,  fecond,  and  laft  terms,  to 
find  the  fum  of  the  feries,  per  Rule  fecond. 

256x4— 4— 1024— 4~  ioio~produft  of  the  fecond 
and  laft   terms,   lefs  the  fquare   of  the  fir  ft  \    then 


difference  divi- 

4 — 2  2 

Jed  by  the  fecond  term  lefs  thejirft~fum  that  the  whole 
cloth  coft. 

BUT  in  finding  the  fum  of  the  feries  by  the  fore 
going  rules,  it  is  neceijjiry  to  have  the  laft  term  giv^- 
en :  therefore  the  next  thing  in  order,  is,  to  Ihew 
how  the  laft  term  of  the  feries,  when  it  is  not  given 
in  the  queftion,  may  be  obtained. 

PROBLEM 


(       '45       ) 


PROBLEM    II. 

The  fir  ft  term  >  Ratio  >  and  number  of  terms  of  a  Geo 
metrical  feries  being  given,  to  find  the  laft  term. 

I.  WHEN  the  firft  term  and  Ratio  are  alike. 

RULE     I. 

1.  WRITE  down  an  Arithmetical  feries  of  a  con 
venient  number  of  terms,  whofe  firit  term,  and  com 
mon  difference  is  unity  or  i. 

2.  WRITE  a  few  of  the  leading  terms  of  the  Geo 
metrical  feries,  under  the  firft  terms  of  the  Arithme 
tical  one. 

Thus     IT>  2>  3>  4>    5'  Indices,  or  exponents. 
I  2,  4,'  8,   1 6,  32,  Geometrical  feries. 

3.  ADD  together  any  two  of  the  indices,  and  mul 
tiply  the  terms  in  the  Geometrical  feries,  which  be 
long  to  thofe  indices,  together,  and  their  produ-ft  will 
be  that  term  of  the   Geometrical  feries,   which  the 
fum  of  thofe  two  correfponding  indices  point  out. 

4.  CONTINUE  the  addition  of  the  indices,  and  mul 
tiply  their  correfponding  terms,  of  the  Geometrical 
feries,  refpectively  as  before,  until  the  fum  of  the  in 
dices  is  equal  to  the  number  of  terms,  the  product 
anfwering  thereunto,  will  be  the  lad  term  required. 

II.  WHEN  the  firft  term  is  either  greater  or  lefs 
than  the  Ratio,  (unity  excepted.) 

RULE    II. 

i,  WRITE  down  an  Arithmetical  feries,  beginning 
with  a  cypher,  the  common  difference,  the  fame  as 
in  the  laft  rule, 

T  £, 


2.  PLACE  the  leading  terms  of  the  Geometrical 
feries,  under  the  Arithmetical,  fd  that  the  cypher 
may  ftand  over  the  firft  term  of  the  Geometrical  fe- 
ries ;  then  add  the  indices,  and  multiply  their  corref- 
ponding  terms  as  before. 

3.  DIVIDE  that  product  by  the  firft  term,  and  the 
quotient  will  be  that  term  of  the  feries,  which  is  de 
nominated  by  the  fum  of  thofc  indices  :  The  reft 
the  fame  as  before. 

III.  WHEN  the  firft  term  is  unity  or  i. 

R  U  L  E    HI. 

WVRTTE  down  the  terms,  and  place  their  indices  as 
in  the  lad  rule  ;  then  add  the  indices,  and  multiply 
the  terms  which  they  denominate,  together,  till  the 
fum  of  the  indices  is  one  lefs  than  the  number  of 
terms,  and  the  refult  will  be  the  iail  term,  as  requir 
ed. 

AN  example  in  each  of  the  foregoing  rules,  will 
make  their  application  eafy. 

In  a  Geometrical  feries,  there  is  given,  t«he  firft 
term=2,  Ratio  2,  and  number  of  terms  12,  to  find 
the  laft  term,  per  rule  i . 

OPERATION. 

Thus,  I  x>  2>  3>  4>    5>    6>  Indices. 
I  2>  4,  8,   16,32,64,  Tf. 

Here,  4+2=6,  the  index  ofthefixtb  term  ;  confe- 
quently  4x16=64,  the  fix  th  term.  Againy  6+6—12, 
tffltf7  64X64=4096^/^4^  term*  as  required. 

Suppofe  the  firft  term  of  a  feries  in  ~,  is  3,  Ratio 
2,  and  number  of  terms  15  i  required  the  laft  term, 
per  rule  2. 

OPERATION. 


(       147      ) 
OPERATION. 


irft      I  °3   l>  2)     3>     4>    S>  Indices. 
>     I  3>  6,  12,  24,  48,  9^>  #v' 


Firft 


24X96=23045  therefore, 
3—  l6%—  eighth  term.      Againy  34-4=7,  and 
;    therefore, 


term.  Z^/y,  7  +  8  —  1  5  ;  whence^  -4    ./.  -------  98304 

O 

—  :  i  $tby  and  laft  term  which  'was  to  be  done. 

Given  firft  term'—  i,  Ratio  4,  and  number  of  terms 
••II,  to  find  the  laft  term,  per  rule  3. 

)0  OPERATION. 

TVm      I  °>   x»  >2    3>    4>  Indices. 

hUS>  1  i,  4,  16,  64,  256,  *. 
Then,  4-}-3  +  3=io~»^w^r  ^  /^rwj  lefs  one— 
index   to  the   nth  term  ;  therefore,  256X^4X64— 
1048576^111^  term  as  was  required. 


Mifcellaneous  £>ueftions. 

A  MAN  hired  himfelf  to  a  farmer,  for  28  weeks 
upon  thefe  confiderations  ;  that  for  the  fir  ft  week  to 
have  i  ct.  ;  for  the  fecond  2  cts.  ;  and  the  third 
4  tfj.  ;  and  fo  on,  in  4f  '-  What  did  his  28  weeks 
wages  amount  to  ? 

The  laft  term  by  the  foregoing  rules,  is,  1 342 17728, 
which  multiplied  with  the  Ratio  (2)  produces 

268435456;  therefore,  - — 43545   — T  —268435455 
cts. "2684354  dol.  $$cts.  the  anfwer.  A 


AnJ.  { 


(       14*       ) 

A  MAN  bought  20  yards  of  velvet,  at  the  follow 
ing  prices,  viz.  for  the  firft  yard  he  gave  i.cts.  -,  for 
the  fecond,  4  cfs.  j  for  the  third,  8  cts.  and  fo  on,  in 
Geometrical  Proportion :  How  much  did  the 
whole  coft  ? 

Anfwer.     20971  del.  50  cts. 

A  MERCHANT  fold  24  yards  of  lace  5  the  firft  yard 
for  3  pins,  the  fecond  for  9,  the  third  for  27  ;  and  fo 
on,  in  triple  Proportion  Geometrical  :  Now  fuppofe 
he  afterwards  fold  his  pins  1 10  for  a  cent  :  What  did 
his  lace  amount  to,  and  what  was  his  gain  in  the 
whole,  when  he  gave  50  cts.  per  yard  for  his  lace  ? 

Lace  corns  toy  4236443047  dot.  20  cts. 
Gain  in  ths  whole,  4236443035  dol.  20  cts. 

A  THRESHER  agreed  with  a  farmer  to  work  fer 
him  25  days,  for  no  other  confideration  than  2  barley 
corns  for  the  firft  day  8  ;  for  the  fecond  32  ;  for  the 
third  -3  and  fo  on,  in  quadruple  proportion  Geometri 
cal  :  How  much  did  his  wages  amount  to,  allowing 
7680  barley-corns  to  make  one  pint,  and  the  barley 
to  be  fold  for  25  cts.  per  bufhel  ? 

Anfwsr.     381774870  dol.  75  ~cts. 

SUPPOSE  a  wheat-corn  had  been  fowed  at  the  crea 
tion,  and  continued  to  increafe  in  a  ten -fold  propor 
tion  every  year,  down  to  the  prefent  time  5  now  al 
lowing  5003  years  for  the  elapfe  of  time:  What 
would  be  the  number  of  wheat-corns  produced  ? 

Here  the  firft  term  being  i,  the  Ratio  10,  and 
the  number  of  terms  5000,  it  is  therefore  plain,  that 
the  laft  term  will  be  i,  haying  as  many  cyphers  an 
nexed,  as  there  are  number  of  terms,  ;le-fs  one  -,  con- 
fequently  its  value  is  1(4999)0^,  where  the  numeral 
figures  included  in  the  parenthefis,  exprefs  the 
number  of  cyphers  annexed  to  the  i  ;  Next  to  find 
the  fum  of  the  feries.  Firft 3 


(      '49      ) 

Firjl,  1(4999)0^X10—1(5000)  tfsytben  1(5000) 
oV— i  =(5000)  9'j  ~  /£<?•  number  cf  y's  therefore 

5OOQ  9>J  , , 

•  •  ••    ' — ^:i 1 1 ii in i in 1 1 iniii   ejv.   to    cooo 

10— I 

f  laces  of  figures— number  of  wheat-corns  produced  j 
which  number  far  exceeds  all  human  imagination  ; 
for  the  whole  fpace  occupied  by  our  folar  fyitem, 
which  is  at  lead  twenty  thoufand  million  of  miles  in 
diameter,  is  by'mtich  too  fmall,  to  contain  the  afore- 
faid  quantity  of  wheat:  Nay,  fuch  a  quantity  would 
take  up  more  fpace,  than  is  contained  in  the  whole 
heavens  on  this  fide  the  fixed  ftars.  Hence  we  may 
learn  the  great  power  of  progreflive  numbers,  and  that 
fmall  portion  of  fpace,  neceffary  to  exprefs  a  number 
by  the  help  of  numeral  figures  contrived  for  that 
purpofe,  which  Ib  far  exceeds  all  our  imagination*. 


CHAP.      II. 

DISJUNCT    PROPORTION, 

OR 
fb*    RULE    of    THREE. 

WH  E  N  of  four  numbers,  the  firft  has  tha 
fame  Ratio  to  the  fecond,  as  the  third  has  to 
the  fourth  :  .Or  when  the  fecond  is  the  fame  multiple 
or  quotient  of  the  firft,  as  the  fourth  is  of  the  third  5 
then  are  thofe  numbers  faid  to  be  in  Disjunct  Pro 
portion. 

IF  four  numbers  are  proportional  diree~lly,   as  the 
firft  to  the  fecond  j  fo  is  the  third  to  the  fourth  ;  then 
will  they  alfo  be  proportional  ;  Inverfely,  Alternate 
ly, 


ly,  Compoundedly,  Dividedly,  and  Mixtly.     (Vid 
Book  ii.  Cbaf.  xrr.) 

SECT.      I. 

DIRECT    PROPORTION, 

OR 
The    RULE    of   THREE    DIRECT. 

THIS  is  fomctimes  called  the  golden  rule,  from  the 
great  benefit  people  in  all  kinds  of  bufmefs  receive 
from  it,  as  well  the  farmer  and  mechanic  as  the  mer 
chant,  &c.  It  confifts  of  four  numbers,  which  are 
proportional,  as  the  firft  to  the  fecond  -,  fo  is  the  third 
to  the  fourth,  as  above  :  The  two  firft  are  a  fuppofi- 
tion,  the  third  a  demand,  and  the  fourth  the  anfwer. 
The  two  fuppofitions  and  the  demand  are  always  giv 
en,  and  the  fourth  required. 

Let  the  four  numbers  be,  a,  £,  c,  d.  Then  a  \  b  :: 
c  \  dy  dire&ly  ;  therefore,  ay^d—by^c,  or  ad~bc,  per 
Lemma  ir,  of  the  laft  Scclion. 

Whence  by  the  nature  of  divifion  bc—a^zd,  that  is, 
if  the  product  of  the  fecond  and  third  terms,  be  di 
vided  by  the  firft,  the  quotient  will  be  the  fourth.  Or 
iince  the  Ratio  of  the  firft  to  the  fecond,  is  the  fame 
ss  that  of  the  third  to  the  fourth  ;  it  follows,  that 
'b~ay,c~d,  that  is,  if  the  fecond  term  be  divided  by 
the  iirft,  and  that  quotient  multiplied  into  the  third, 
it  will  produce  the  fourth. 

Now,  in  order  to  prepare  your  numbers  for  obtain 
ing  a  fourth  proportional,  according  to  the  forego^- 
ing  rules,  you  inuft  obferve  the  following 

RULE. 


R  U  L  E. 

WRITE  that  number  which  is  of  the  fame  name 
with  the  number  fought,  in  the  middle  place,  and 
the  other  two  fo,  that  the  exprefllon  may  read  accord 
ing  to  the  nature  of  the  queftion. 

Let  the  following  conditions  be  exprefied  in  num 
bers. 

What  is  the  coft  of  24lb.  of  cheefe,  when  the 
price  of  3lb.  is  20  cfs.  ? 

Here  the  middle  number  muft  be  coft,  becaufe 
the  fourth,  or  number  required,  is  always  of  the 
fame  name  and  denomination  of  the  fecond,  by  the 
nature  of  the  proportion  :  Hence  the  above  condi 
tions  in  numbers,  is, 

Thus,  jib.  zocts.  24bl. ;  that  is,  if  3  pounds  coft 
so  cts.  what  will  24  pounds  coft  ?  Then  to  find  a 
fourth  number,  proceed  as  before  directed. 

Note.  If  the  firft  and  third  numbers  are  not  &f  the 
Jame  name,  they  muft  be  made  Jo  by  the  rules  of 
reduction  :  Alfo,  if  any  of  the  numbers  are  com 
pounds,  they  mujl  be  reduced  to  the  leafl  dejivmin- 
ation  mentioned. 

EXAMPLES. 

If  4lb.  of  cheefe  coft  32  cts. ;  what  will  32olb«  coft 
ic  fame  rate  ? 


OPERATION 


(      '5*     > 


OPERATION. 

Thefe  numbers  being  placed  according  to  the 

\b,  cts.     Ib. 
rule,    will  ftand  thus,   4  :  32  ::  320 

3* 

640 
960 

4)  ,10240 

i (oo)  25(60=125  dol.    60 
[cts.  the  anjwer. 

Or,  32-7-4^:8  ;  therefore)  320X8  —  2560  ^.  —  25 
Jol*  60  els.  the  fame  as  before. 

What  will  6  yards  of  holland  coft,  when  the  price 
of  40  yards,  is  24  dol.  40  cts.  ? 

OPERATION. 

yd.  dol.  cts.   yd. 
As  40  :  24  40  ::  6  ftated. 

^heny   24.4O-4-4On=.6i,  and  6 X. 6 1=366  cts.^ 
3  dol.  66  cts.  the  anfwer. 

Find  the  value  of  loolb.  of  flax,  when  the  price  of 
lib.  is  12  cts  ? 


OPERATION; 


Ib.  cts.       Ib. 
As  i  :  12  ::  106 


iz  1 2  dol.  the  anjwcr. 

What 


(      153      ) 

What  is  the  coft  of4olb.  of  cheefe,  when  the 
price  of  jlb.^is  15  cts. 

OPERATION. 

Firft,  15-7-3  —  5,  the  ratio  of  the  firft  term  to  the 
\fecond. 

Then,  40X5  —  200  cts.~i  dol.  the  anfwer. 

What  is  the  coft  of  Sylb.  of  tobacco,  at  84  cts.  per 
Ib.  ? 

OPERATION. 

Ib.  cts.     cts.      Ib. 
As  i  :  84-  =8.5  ::  87 


435 
696 

739-5=739-1- ^-= 7  ^ 
[39i  cts-  the  anfwer. 

A  goldfmith  fold  a  tankard  for  29  ^/.  97  r/j.  at 
the  rate  of  i  */0/.  n  cts.  per  oz.  :  What  was  the 
weight  of  it  ? 

Anfwer.  '  27  oz. 

A  man  bought  Iheep  at  i  <&/.  1 1  r/j-.  per  head,  to 
the  amount  of  5 1  doL  6  cts.  :  How  many  Iheep  did 
he  buy  ?  Anfwer.  46, 

SECT.       II. 

RECIPROCAL,  or  INSERTED  PROPORTION* 

OR 
*ba    RULE    of   THREE   INDIRECT. 

THIS  kind  of  proportion,  is  the  reverfe  of  the 
former,  as  to  the  performance  j  for  the  greater  the 

U  third 


third  term  is,  in  refpect  of  the  firft,  the  lefs  will  b( 
the  fourth,  in  refpect  of  the  iecond  ;  whereas  in  di- 
red  proportion,  the  greater  or  lefs'  the  third  term  is.) 
in  rcfpect  of  the  firft,  the  greater  or  lefs  will  be  the 
fourth  term,  in  refpect  of  the  fecond  ;  but  to  illuf- 
trate  the  former.  If  two  men  can  produce  a  certain: 
effect  in  12  days  :  In  how  many  days  would  6  mer 
produce  the  fame  ?  Here  it  is  manifeft,  that  6  mer 
would  produce  the  effect  in  lefs  time  than  2  ;  anc 
therefore  the  greater  the  third  term  is,  the  lefs  will 
be  the  fourth.  Again,  if  lomen  can  produce  a  cer 
tain  effect  in  6  days  :  In  how  many  days  would  4 
men  do  the  fame  ?  Here  it  is  evident,  that  10  men 
would  produce  the  effect  in  lefs  time  than  4  men  \ 
and  therefore  the  lefs  the  third  term  is,  the  greater 
will  be  the  fourth  :  Confequently,  more  requires  lefs^ 
and  lefs  requires  more,  in  indirect  proportion. 

HERE  the  fame  rule  is  to  be  obferved,  in  Hating 
your  queftion,  as  in  the  former  proportion,  and  did 
refults  in  refpect  of  names  and  denominations  are  the| 
fame  alfo  :  Then  to  find  a  fourth  proportional,  pro-! 
ceed  with  the  following  rules. 

RULE    I. 

MULTIPLY  the  firft  and  fecond  numbers  together, 
and  divide  that  product  by  the  third  ;  the  quotient 
reiulting  will  be  the  fourth  proportional  required. 

RULE     II. 

DIVIDE  the  fecond  number  by  the  third,  and  that 

2uotient  ^multiplied  into  the  firft,    will  produce  the 
ajiirth, 

RULE 


(      '55      ) 


RULE    III. 

DIVIDE  the  third  term  by  the  firft,  and  the  fecond 
:rm  by  this  quotient  5  and  the  refulting   quotient 
iil  be  the  fourth  number. 

EXAMPLES. 

If  5  men  can  perform  a  certain  piece  of  work  in 
days  :  How  long  will  four  men  be  in  doing  the 


ame 


OPERATION. 


Men.  D.  Men. 
S     8     4 

5_ 

4)  40    D. 


Or, 

4_4Q 

8+r~~ 

mo  days  as  before. 


If  2,0  bufliels  of  grain,  at  50  o?#/.r  per  bulhel,  will 
pay  a  debt  :  How  many  bufhels  at  60  cents  ^r  bufh- 
il  will  pay  the  fame  ? 


OPERATION. 


OPERATION, 
cts.  Bujb.     cts. 

50         20         60 
20 


6(o)lOO(o 
164 

Anfwer.     164  bujhds. 

If  2  yards  of  cloth,  i  yard  and  3  quarters  wide, 
is  fufficient  to  make  a  coat ;  how  many  yards  of  i 
yard  wide,  will  make  the  fame  ? 

OPERATION. 


^ 

3^— 3~  yards  the  anfwer. 

A  man  being  defirous  to  draw  off  a  cafk  of  bran 
dy  into  bottles,  finds  that  if  he  makes  life  of  three 
quart  bottles,  it  will  require  60  :  How  many  five- 
pint  bottles  will  it  require,  to  draw  off  the  aforefaid 
cafk  of  brandy.  Anfwer.  72  bottles. 

A  man  bought  a  piece  of  cloth  9  quarters  wide, 
and  ii  quarters  long  :  How  many  yards  of  3  quar 
ters  cloth  will  line  it  ?  Anfwer.  8.1  yards. 

If  3'-yai*ds  of  yard-wide  cloth  will  make  a  coat : 
How  many  yards  of  7  quarters  cloth,  will  make  the 
?  Anfwer.     2 yards. 

SECT. 


(      157      ) 

S  E  C  T.       III. 
COMPOUNDED      RATIO. 

COMPOUNDED  Ratio  is  when  the  antecedent  and 
confequent  taken  together,  is  compared  to  the  confe- 
quentitlelf :  thus,  a  :  I  ::  c  :  ^,  directly,  therefore  .by 
cornpofition  ;  as  a-\-b  :  b  ::  c-\-d :  d. 

Note.  The  fame  Rule  is  to  be  oljerved  here,  as  in 
direft  proportion. 

EXAMPLES. 

If  A  can  produce  a  certain  effe6t  in  5  days,  B  can 
do  the  fame  in  7  days  ;  fet  them  both  about  it  toge 
ther,  in  what  time  will  it  be  finilhed  ? 

ORERATION. 


12)35(2  days. 
24 


22  bourf.    AnJ.  2  days  lib. 
If  A  in  in  5  hours,  can  make  1000  nails,  B  in  B 
hours,  can  make   2000  :  In  what  time  would  they 
'  jointly  make  50000  nails  ? 

Here 


Here  you  muft  firft  find  in  what  time  each  perfon 
would  make  50000  nails,  and  then  proceed  as  in  the 
lait  example. 

OPERATION. 


n.     h.         n. 


As    icoo  :  5  ::  50000  :  50000X5-^-1000  =z  250 
boars,  the  time  it  would  take  A  to  make  50000  nails. 
n.        h.          n.       _ 

As    2000  :  8  ::  5000  :  50000X^-7-2000^:200 
hours,  the  time  it  would  take  B  to  make  50000  nails. 


e^  as  2504-200  :  200 : :  250  :  200  x  250-— 
450—  1 1  r~  hour s y  the  time  it  would  take  them  jointly 
to  make  50000  nails,  as  was  required. 

Note.  From  this  operation,  we  have  the  following 
general  theorem  forjolving  all  queftions  ofaftmil- 
ar  nature,  let  the  ferfons  or  agents  employed,  be 
any  numler  whatever, 

THEOREM. 

MULTIPLY  the  joint  effect  with  the  time  each  one 
would  produce  his  particular  effect,  and  divide  the 
product  by  the  faid  particular  effect ;  then  multiply 
all  the  refuhing  quotients  together  for  a  dividend, 
and  make  the  fum  of  them  a  divifor  -,  then  divide, 
and  the  refulting  quotient  will  be  the  time  required. 


SECT, 


(      159     ) 

S  EC  T.      IV. 

DIVIDED      RATIO. 

DIVIDED  Ratio  is  when  the  excefs  wherein  the  an 
tecedent  exceeds  the  confequent,  is  compared  with 
the  confequent :  fbus,  a\b\\  c\  d> direftiy  ,  therefore 
by  divifien  as  a — b  :  b : :  c — d :  d. 

EXAMPLES. 

If  A  can  do  a  piece  of  work  in  8  days,  A  and  B 
can  do  it  in  5  days :  In  what  time  can  B  do  the  fame 
work  ? 

OPERATION. 

u.     h. 

As*— 5^3  -.5::  8: 5X8^3-40-3-  '3  8>  ** 
time  required. 

Two  flitps,  one  in  chafe  of  the  other,  the  head- 
moft  (hip  is  48  miles  diftant  from  the  other,  and  fails 
at  the  rate  of  4  miles  per  hour,  and  the  fternmoft 
Ihip  at  the  race  of  7  miles  per  hour :  How  long  be 
fore  the  fternmoit  fhip  will  overtake  the  o:her  ? 

OPERATION. 

As  7 — 4 IT 3  *•  i  : :  48  :  48X1-7-3—16  bcurs,  tbe 
time  required. 

A  hare  is  is  50  leaps  before  a  grey -hound,  and 
takes  4  leaps  to  the  grey-hound's  three  ;  but  2  of 
the  grey-hound's  leaps  are  as  much  as  three  of  the 
hare's  :  How  many  leaps  rnuft  the  ^r?.y-hound  take 
to  catch  the  hare  ? 

Her  £  you  mufl  firft  find  how  many  kafs  of  ths  bart, 
anfivers  to  three  of  the  grey-hound' V  :  Tbu^  2:3^  3 


?hen,  as  4.5—  4~-5  :3:*-5°:3X5o-~-.5—  300 
/fo  anjwer. 

The  hour  and  minute-hand  of  a  clock  are  exactly 
together  at  12  o'clock  j  when  are  they  next  together  I 

Here  the  proportion  of  the  vehfities  of  the  hour  and 
minute-hand^  is  as  i  to  12.  Thereforeyii  —  iHn  :  i 


::  12  t  i2Xi-S-u~ih.  5-/-A  the  anjwer. 

If  A,  B  and  C,  can'  produce  a  certain  effect  in  12 
days,  A  can  do  it  in  30  days  and  C  in  50  days,  in  what 
time  will  B  do  the  fame  work  ? 

Fir  ft  find  the  time  in  which  A  and  C,  would  produce 
the  effeff  jointly,  by  Ratio  of  compofition.  Thus, 


30-1-50  :  50  ::  30  :  50X30-^70  =21!  days, 

as  21^  —  12^:91  :  12::  21  -T  :  i$^~y  the  time  requir 

ed. 

There  is  an  ifland  100  miles  in  circumference, 
and  two  footmen,  A  and  B,  fet  out  together,  to  trav 
el  the  fame  way  round  it,  A  travels  15  miles  per  day, 
and  B  17  miles:  When  will  they  come  together 
again  ? 

Firfty  find  how  many  miles  B  muft  travel  to  over 
take  Ay  after  their  departure  :  T'btts,  as  17  —  15:1:2:  17 
::  100  :  850,  the  number  of  miles  B  muft  travely 
which  is  50  days  journey  ;  therefore  they  will  be  together 
Ggain  50  days  after  their  departure. 

There  is  three  pendulums  of  unequal  lengths  -,  the 
firft  of  which  vibrates  once  in  12  feconds,  the  fecond 
in  1  8  feconds,  and  the  third  in  24  feconds  :  Now 
fuppofing-  them  all  to  move  from  a  line  of  conjunc 
tion,  at  the  fame  moment  of  time  :  When  will  they 
come  into  the  fame  fituation  again,  and  move  on 
together  ?  Firft 


Firft  y  find  the  time  when  the  two  firft  pendulums 
W9ve  on  together  y  as  in  the  loft  example  :  Thus,  1 8— - 
12:  1 8  ::  i  :  i%  ^1+6^ $ytbfmmber  of  vibrations  $ 
the  firft  y  which  is  performed  in  36  Jeconds~i  vibra~ 
tions  of  thefecond.  ThereforCy  after  th?  firft  has  vi 
brated  3  timeSy  and  the  fecond  2,  they  will  move  on  to 
gether  again. 

In  the  next  placey  we  muft  examine  into  the  fituatim 
of  the  third  pendulum ,  at  the  conjunction  of  the  two  firft. 
In  36  fecondsy  there  is  1.5  vibration  of  the  third  pen~ 
dulumy  which  is  thereforey  .5  of  a  vibration ,  diftant 
from  the  conjunction  of  the  other  two  ;  whereforey  .5:1 
::  3  :  6,  the  number  of  vibrations  of  the  fir  ft y  at  which 
time,  they  all  come  into  a  line  of  conjunct  ion  y  and  move  on 
together.  Conjequentlyy  when  the  firft  has  made  6  vi- 
brationSy  the  Jecond  will  have  performed  4,  and  the 
third three^zi^y.3=?]ijecondsy  the  time  required. 

If  A  can  do  a  piece  of  work  in  20  days  ;  A  and 
B  in  13  days ;  A  and  C  in  1 1  days ;  and  B  and  C 
in  LO  days  :  How  many  days  will  it  take  each  perfon 
i  to  perform  the  fame  work  ? 

OPERATION. 

As  20- — 13  : 13  ::  20 :  374-  the  time  that  B  would 
1  do  it. 

As  20— ii  :  ii  ::  20 :  244  the  time  that  C  would 
do  it, 


C  HA 


CHAP.      III. 

SIMPLE      INTEREST. 

SIMPLE  intereft  is  a  premium  of  a  certain  fum 
paid  for  the  loan  of  money  borrowed  for  a  par 
ticular  term  of  time,  at  any  rate  per  cent  or  hundred, 
as  the  borrower  and  lender  fh all  agree. 

THUS,  if  100  dollars  be  lent  at  6  per  cent  per  an 
num,  the  premium  for  i  year  will  be  6  dollars,  for 
a  years  12  dollars,  for  3  years  18  dollars;  and  fo 
on. 

THE  fum  lent  is  called  the  principal,  and  the  pre 
mium  per  100,  the  Ratio  or  rate  per  cent  -,  and  the 
amount  is  the  principal  and  intereft  added  together. 

ALL  the  varieties  of  fimple  intereft,  are  comprifed 
in  the  following  cafes. 

C  AS  E  I. 

When  the  fum  lent,  Is  for  any  number  of  years  >  and 
the  rate  per  cent,  any  number  of  dollars. 

RULE. 

MULTIPLY  the  principal  with  the  number  of  years, 
and  that  product  with"  the  Ratio,  and  divide  by  1005 
the  quotient  refulting,  will  be  the  intereft  required. 

EXAMPLES. 

Required  the  intereft  of  700  dollars,  for  4  years, 
at  6  per  cent  per  annum  ? 

OPERATION. 


OPERATION. 

700 
4 

2800 
6 

1(00)168)00 
'nfwer.     168  dollars  y  the  inter  eft  required. 

Required  the  intereft  of  3520  dollars,  for  7  years, 
at  6  per  cent  per  annum. 

OPERATION. 


1(00)1478(40=1478  dol.    4octs.    the 

\anjwer. 

What  is  the  intereft  of  57821  dollars,  for  5  years, 

at  5  per  cent  per  annum  ?  AnJ.  2891  dol.  5  cts. 

What  is  the  intereft  of  5972  dollars,  for  12  years, 

at  3  per  cent  per  annum  ?  AnJ.  716  dol.  64  cts. 

•CASE      II. 

"  When  thefum  is  lent  for  years  and  months  5  the  Ra 
tio  the  fame  as  before. 

RULE. 

REDUCE  the  number  of  months  into  the  decimal  of 
a  year,  then  multiply  the  principal  with   the  time, 

and 


and  that  product  with  the  Ratio,  then  divide  by  100 
and  you  will  have  the  intereft  required. 

.  Or>. 

MULTIPLY  the  principal  with  the  number  of  years, 

and  take  parts  of  the  principal  for  the  reft  part  of 
the  time,  a^.d  add  them  to  the  reft  $  then  proceed  as 
before  directed. 

Required  the  intereft  of  735  dollars,  for  5  years/ 
4  months,  at  5  per  cent  per  annum. 

OPERATION. 

4  monthszr-J- of  a  year,  3)735 

5 

3675 

245=735-^3 

3920 

5   ratio. 

1(00)196)00=196  dollars,  the 
[intereft  required. 

Required  the  intereft  of  52374  dollars,  for  7  years 
8  months,  at  6  per  cent  per  annum. 


OPERATION. 


OPERATION. 

*?  of  a  year,  3)52374 
7 


366618 

i745%-±  *f  5*374 
17458 


401534 


i(oo)-24°9a)°4— 24092    4,  the  in- 
\tweft  required. 

What  is  the  intereft  of  32104  dollars,  for  4  years, 
3.  months,  at  5  per  cent  per  annum  ? 
AnJ.  6827  dol.  10  cts. 

CASE     III. 

When  the  Rath  is  dollars  and  farts  of  a   dollar* 
the  reft  the  fame  as *  before. 

RULE. 

1.  REDUCE  the  number  of  months  into  the  decimal 
of  a  year,  and  multiply  the  principal  with  the  whole 

time. 

2.  REDUCE  the  fractional  parts  of  the  Ratio   into 

the  decimal  of  a  dollar. 

3.  MULTIPLY   the  former  refult  with  the  latter, 
and  divide  by  100,  and  you  will  have  the  intereft  re 
quired  :  Or, 

MULTIPLY  the  principal  with  the  number  of  years, 
and  take  parts  of  the  principal  for  the  reft  part  of  the 
time,  and  add  them  to  the  former  product  •>  then 

multiply 


(      166      ) 

multiply  this  product  with  the  dollar's  part  of  the 
rate,  and  take  parts  of  the  multiplicand  for  the  reft 
part  of  the  rate,  and  add  them  to  the  latter  product  j 
then  divide  them  by  100,  and  you  will  have  the  in- 
tereft  required. 

EXAMPLES. 

Required  the  intereft  of  700  dollars,   for  3  years 
6  months,  at  6~  per  cent  per  annum. 

OPERATION. 

700 


3500 

2IOO 
245O.O 

6,$  ~ 


122500 
147000 

doL     cts. 


1(00)159)25.00—159     25,  the  anfwer* 
Or.     6  months—^  a  year  2)700 

3 


2IOO 


for  tbt  -per  cent  2)2450 

6 


14700 
1225 

doL  cts. 


I(°°)I59(25=I59  25  as 
/"'*•  Required 


i7 

Required  the  intereft  of  3520  dollars  17  cents,  for 
2  years  6  months,  at  5^  per  cent. 

OPERATION. 
2)3520.17 


704034 
176003.5-3520.17-7-2 


4)8800.375 
5 

44001875 
2200.093 

doL  cts. 


1(00)462(01.968—462  i.   968    tie 

\anf. 
CASE     IV. 

When  fhefum  is  lent  for  any  number  of  weeks. 

RULE. 

REDUCE  the  number  of  weeks  into  the  decimal  of 
a  year,  and  proceed  as  in  the  laft  cafe. 

Or, 

FIND  the  intereft  of  the  given  fum,  according  to 
the  foregoing  rules  for  one  year  $  then  fay,  as  52,  the 
number  of  weeks  in  a  year,  is  to  the  intereft  thus 
found  ;  fo  is  the  given  number  of  weeks,  to  the  in 
tereft  required. 

EXAMPLES. 

Required  the  intereft  of  720  dollars,  for  10  weeks, 

at  5-i  per  cent  per  annum, 

OPERATION. 


OPERATION. 

720 

.19211:  time nearly. 


138.240 


691200 
691200 


dol.  cts. 


7,60.3200:1:7     60/32  the  anfwer* 

ix).  dol.  cts.     w. 
Or,  720    As  52:39  60  ::  10 

*       5-5 


10 


3600 
3600 


52\396-°°(7-6*~7  Jol.  61  fts. 


J  20 

312 


80 


28 

Note.  Tfareafw  why  the  two  methods  of  operation  a- 
loroe>do  not  bring  cut  tbefameanjwer,  is  lecauje  the 
decimal  of  10  weeks  fan  never  be  exaftly  found  ; 
yet  the  err  GUT  artfing  from  any  Juch  computation, 
will  be  inconfidcrabh. 

Required 


I.         LW7        / 

dol.  cts. 

Required  the  intereft  of  5  27   2,  for  13  weeks,   at 
;i  per  cent  per  annum. 


659000 
659000 

i  (00)7  2(49*000=7*  dol.  46  cts.  the  anf. 

C  A  S  E     V. 

When  tbefum  is  lent  for  any  number  of  days. 
RULE. 

REDUCE  the  days  into  the  decimal  of  a  year,  and 

proceed  as  in  the  laft  cafe. 

Or, 

T  MULTIPLY  the  given  fum  with  the  number  of 
dap,  S  US  Produagwith  theRatio £  a  dmdena 

V  MULTIPLY  365,  the  number  of  days  m  a  year, 
with  ,00  for  a  divifor  ;  then  divide,  and  the  quotient 
will  be  the  intereft  required. 

As  365  days,  is  to  the  intereft  of  die  principal  for 
gjjff  « 


EXAMPLES. 

Required  the  intereft  of  300  dollars,  for  219  days 
at  6  per  cent  per  annum. 

OPERATION. 

Or,  219  365 

3°°  .  300  100 

65700       36500 
180.0  6 

6=ratioy  .     doL  cts. 

d.  365(00)3942(00)  10 80  aslefin 

1(00)10)80,0=1080^.^.365 

292.0 
2920 

o 

Required  the  intereft  of  1000  dollars,  for  35  days 
at  6  per  cent  per  annum. 

OPERATION. 
d.      dol.     d. 

1000          As  365:  60::  35 

60 

6o-°°  365x2100  (5  del.  75  cts. 

71825 

275.0 


15150 

1825 
dot.  cts. 


125 

CASE 


CASE     VI. 

Wbtn  the  principal,  Rath,  and  intereft  are  given  t* 

id  tke  time. 

RULE. 

i.  FIND  the  intereft  of  the  principal  for  one  year, 

it  the  eiven  rate. 
2    SAY  as  the  intereft  thus  found,  is  to  one  year  ; 

b  is  the  given  intereft,  to  the  time  required. 

EXAMPLES. 

Required  the  time  in  which  500  dollars  will  gain 
150  dollars,  at  6  per  cent  per  annum. 
OPERATION. 

dol.     y.       J°I- 
As  30:1::  15° 
6  * 


Find  in  what  time  700  dollars  will  gain  159    o. 
cts.  at  6iper  cent  per  annum. 

OPERATION. 

700  del.  cts.  y.    dot.  cts 

A  45  5o:i: 


f(  172  ) 

• **-  dol.  cts. 

Required  the  time  in  which  283  334.  will  amount 
to  370  dol.  50  cts.  at  6  per  cent  per  annum. 

OPERATION. 

dol.  cts.  dol.  y.  dol.  cts. 

33r  As  17  i  87   i6£ 

6  i 


17.0000 


17X3=51)261. 50(5,12^*™,  the 
25S  {anfwer* 


140 
1 02 


CASE     VII. 

When  the  Ratio,  time,  and  amount  are  riven  to 
find  the  principal. 

RULE. 

As  the  amount  of  100  dollars,  a*  the  rate  per 
:cnt  and  time  given,  is  to  100  dollars ;  fo  is  the  giv 
en  amount,  to  the  principal  required. 

EXAMPLES. 

9  Required  the  principal  that  will  amount  to  7766 
dot.  40  cts.  in  7  years,  at  6  per  cent  per  annum. 

OPERATION, 


(      173      ) 


OPERATION. 

TOO 

6 

6.00 
7 

42.00 
doL  dol.  cts.     dol.  cts. 

As  100+421:142 : 100 ::  3766  40  :  2793  23  \\\ 

the  anfwer. 

Required  the  principal  that  will  amount  to  861 
dollars  in  4  years,  at  6  per  cent  per  annum. 

OPERATION. 

100  100+24^124: 100  ::  868 

4  ioo 

400  124)86800(700 

6  868 

24.00  ooo 

The r efcr  e  700  doL  is  the  principal  required. 

Requkred  the  principal  that  will  amount  to  270 
dollars,  in  2  years  at  6  per  cent  per  annum. 


OPERATION. 


(      '74      ) 


OPERATION. 
100    As  100+12=:  1  12 

2 

200 

6 


12.00 


16 

)  241  doL  7T|4  cts.  is  the  principal  re 
quired. 

Admit  I  have  a  legacy  of  196  dol.  66-*  cts.  to  pay, 
but  is  not  due  till  the  end  of  3  years,  and  the  lega 
tee  being  in  want  of  money,  defires  I  would  lend  him 
fome  :  What  fum  muft  he  have  to  amount  to  his 
legacy  in  3  years,  at  6  per  cent  per  annum  ? 

Anfwtr.     1  66  dol.  66|-  cts. 

CASE     VIII. 

When  the  principal,  amount  >  and  time  are  given  to 
find  the  Ratio. 

R  ULE. 

i.  SUBTRACT  the  principal  from  the  amount,  and 
the  remainder  is  the  intereft. 

2. 


(      175     ) 

2.  SAY  as  the  given  principal,  is  to  its  intereft ;  fo 
is  100  dollars,  to  the  intereft  of  100  dollars  for  the 
given  time. 

3.  DIVIDE  the  intereft  of  i  oo  dollars  thus  found 
by  the  given  time,  and  the  quotient  will  be  the  ra 
tio  required. 

EXAMPLES. 

Required  the  rate  per  cent  per  annum  fuch,  that 
1240  dollars  may  amount  to  1400  in  3  years. 

OPERATION. 

1400   1240:  200  ::  ico 
i 200         100 

200   1 24)0}  2ooo(o(  1 6. 1 2 

/I24 

760 

744 

16.0 

124 

360 

248 

dol.  cts.  112 
— 5.37  the  Ratio  required. 


Required 


Required  the  rate  per  cent  per  annum,  that  100 
dollars  in  7  years  will  amount  to  135  dollars. 

OPERATION. 

135      As  100  :  35  ::  100 
100  100 


3$=intcreft,  1)00(00(35 

7 (35  </<?/. 

$~  ratio  req. 

At  what  Ratio  will  3333  dollars  33  J-  cents  amount 
to  4000  dollars  in  20  years.  Anfwer*  6  dot. 

CASE    IX. 
COMMISSION    or    PROVISION. 

THIS  is  a  premium  allowed  to  faftors  for  buying 
or  felling  goods,  wares,  or  merchandize,  at  fo  much 
per  cent,  without  any  regard  to  time  -,  which  rate  is 
governed  according  to  the  cuftoms  of  particular 
places. 

THE  method  of  proceeding,  is  the  fame  as  in  cafe 
in,  except  no  regard  is  had  to  time. 

EXAMPLES. 

If  I  buy  goods  for  my  correfpondent  in  Philadel 
phia,  to  the  value  of  4000  dollars  :  What  may  I 
demand  for  my  commifTion,  at  44-  per  cent  ? 


OPERATION. 


OPERATION, 

4000 


.  thsanfwer. 

Required  the  commiflion  for  felling  5720  dollars 
worth  of  goods,  at  2~  per  cenr. 

OPERATION. 


143*000=143  doL  the  anf. 

My  correfpOhdent  fends  me  word,  that  he  has  dif- 
burfed  goods  on  my  account,  to  the  value  of  13333 
dollars  334.  cents  :  What  is  his  commiflion  at  ai  per 
cent  ?  Anfwer.  333  doL  33rCts* 

CASE    X. 

BROKERAGE. 

BROKERAGE  is  an  allowance  of  fo  much  per  cent, 
made  to  perfoas  called  brokers,  for  finding  cuftom- 
ersa  and  felling  to  them  goods,  wares,  &c,  -which  be 
long  to  qther  men^ 

Z 


RULE. 

FIND  the  intereft  of  the  given  fum,  at  one  peri 
cent  i  or  which  is  the  fame  thing  ;  divide  the  given 
fum  by  100,  and  take  parts  of  the  quotient,  agree 
ing  with  the  rate  per  cent. 

Or, 

REDUCE  the  rate  per  cent  to  a  decimal,  and  mul 
tiply  it  with  the  given  fum  -,  then  divide  by  100,  and 
the  quotient  will  be  the  anfwer- 

EXAMPLES. 

Required  the  Brokerage  of  1000  dollars,  at  25  cents 
per  cent. 

OPERATION. 

1(00)10(00 

25  cents—  ^  of  a  dollar,  therefore,  10-7-4=2  dollars 
50  cents  =  Brokerage  of  1000  dollars  divided  by  4:= 
.Brokerage  required. 

Or, 

1000 


2.5000—2  dol.  50  cts.  as  be- 

[fore. 

Required  the  Brokerage  of  324  dollars  40  cents, 
4  of  a  dollar  per  cent. 

OPERATION. 


(      '79     ) 


OPERATION. 

324.40 

.  20=}  of  a  dollar > 


1(00)64.8800^:64.88  cts.   t he  Brokerage 
'jquired. 

What  is  the  Brokerage  of  15600  dollars,  at  77 
:ents  per  cent  ?  AnJ.  120  doL  12  cts. 

CHAP.      IV. 

COMPOUND    INTEREST. 

GO  M  P  O  U  N  p. Intereft  arifes  from  the  com 
putation  of  the  intereft  of  any  principal  added 
,:o  its  intereft,  when  the  payment  Jhould  be  made  ; 
vhich  forms  a  new  principal  at  every  time  when 
I  Repayments  become  due;  and  is  for  this  realbn, 
bmetimes  called  intereft  upon  intereft. 

THUS,  if  100  dollars  be  put  to  intereft  at  6  dollars 
)er  cent  per  annum  ;  at  the  end  pf  the  firft  year,  the 
ntereft  will  be  6  dollars  as  in  fimple  intereft,  which 
f  added  to  its  principal  will  be  106  dollars,  for  a  new 
)rincipal  the  fecond  year,  which  principal  at  the 
:nd  of  the  fecond  year,  will  amount  to  112  dol- 
ars  36  cents  ;  which  is  36  cents  more  than  if  100 
lollars  had  been  put  out  at  fimple  intereft  only. 

THE  Compound  Intereft  of  any  fum  may  be  found 
)y  the  following 

RULE. 

i.  FIND  the  intereft  of  thepropofed  fum  for  the 
irftyearatthe  given  rate  per  cent,  as  in  fimple  in- 
srcft,  2. 


2.  ADD  this  intereft  to  its  principal,  which  amount 
makes  the  principal  for  the  fecond  year. 

3.  FIND  the  intereft  of  the  fecorid  year's  principal,, 
in  the  fame  manner  as  \ou  did  the  firft,  and  add  it  to! 
its  principal,  for  the  third  year's  principal*  which 
mull  be  computed  as  before ;  and  fo  on,  for  the  time 
required . 

4.  SUBTRACT  the  given  principal  from  tfye   laft 
amount,  and  the  remainder  will  be  the  Compound' 
Intereft  required.         Or, 

!FIND  the  amount  of  one  dollar  for  one  year,  at 
the  given  rate  per  cent,  and  multiply  it  continually 
with  the  principal,  as  many  times  as  the  given  num 
ber  of  years,  and  the  refill  ting  product  will  be  the 
amount ;  from  which  fubtradfc  the  principal,  and  the 
remainder  will  be  the  Compound  Intereft. 

EXAMPLES. 

"Required  the  Compound  Intereft  of  100  dollars, 
for  3  years,  at  6  per  cent  per  annum. 

OPERATION 

tOO  IOO 

6  6 

6.00  106 

6 

6.36  6.7416 

,  119  dpi,  i o.  1 6  cts. — i oo  doL  —19  d*l.  i o.  1 6 

intereft  require-d. 

Or, 


106 
6.36 

112.36 
6.7416 

•112.36 

6 

119.1016 

:  106  ::  i  :  i.o6f  iooxi.o6Xi.o6X 
the  amount  of  i  dollar  for  i<  t;.<p.6'^:  119.1016=11 9 
year>  at 6 per  cent.  [dol.  10.16  cts. 

fhen,  119  dol.  10.16  cts. — IQQ doL  —  iydol.  10.16 
cts.  the  fame  as  before. 

THE  following  is  a  Table  of  the  amount  of  i  dollar, 
from  i  to  30  years  ;  for  the  more  ready  computing 
Compound  Intereft.at  6  per  cent  per  annum. 


J? 

& 
"t 

The  amount  of  j 
dol.  at  6  pgr  cent, 

*3c  .  comp  .  inter  eft  . 

Is 

y  ^  amount  of  I 
</o/.  at  6r  per  cent, 
&c  .  comp  .  inter  eft  . 

? 

•I  he  amount  of  \ 
dol.  at  6  per  cent, 

&c.ccmp.intereft. 

I 

.06 

II 

1.898298558 

21 

3-3995636oO 

2 

.1236 

12 

2.012196471 

22 

3-603537416 

3 

.I9IOl6 

*3 

2.132928260 

23 

3.819749661 

4 

.26247696 

H 

2.260903955 

24 

4.048934641 

5 

.338225577 

*5 

2.396558193 

25 

4.291370719 

6 

.4185191  12 

16 

2.540351684 

26 

4.549382962 

7 

.503630259 

*1 

2.692772785 

27 

4.822345940 

8 

.593848074 

18 

2-8.54339I52 

28 

5.111686697 

9 

.689478959 

T9 

3.025599502 

29 

5.418387899 

10 

.790847696 

20 

3.207135472 

jo 

5-74349I?29 

?v  the  above  table,  the  amount  of  any  fum  may 
be  computed  from  i  to  30  years,  by  only  multiply 
ing  the  principal  with  the  numbers  {landing  againft 
the  number  of  years  in  the  table,  and  the  product 
will  be  the  amount  required. 

EXAMPLES. 

Required  the  amount  of  127  dollars,  for  7  years,  at 
6  per  cent  per  ^nnum. 

OPERATION, 


OPERATION. 

dgainft  7  in  the  table  is  1.503630 

127 


10525410 

3007260 

1503630 


~I90  96.1 


,  Required  the  Compound  Intereft  of  555  dollars, 
for  30  years,  at  6  per  cent  per  annum, 

OPERATION. 

dgainft  30  ^  /^  tails  is  5.743491 

,555 


28717455 
28717455 

28717455 

3187.63750511:  amount. 

<?ben,  3187    dol.    63,7505  f/j.  — 555  doL  —2632 
doL  63.7505^.?.  the  intereft  required. 


CHAP. 


C  H  A  P.       V. 

REBATE    or    DISCOUNT. 

REBATE  or  Difcount  is  when  any  fum  of  mo 
ney  is  due  at  a  certain  time  to  come,    and  the 
debtor  is  ready  to  make  prefent  payment,  provided 
he  can  have  allowance  made  him  at  a  certain  rate  per 
cent  per  annum,  which  allowance  is  called  the  Rebate 
or  Difcount,   and  the  prefent  payment,  a  fum  of 
money,  which  ifputtointereft,  would  amount  to  the 
given  fum,  at  the  rate  per  cent  and  time  given. 
THE  Rebate  of  any  fum  is  found  by  the  following 

RULE. 

As  the  amount  of  100  dollars,  at  the  rate  per  cent 
and  time  given,  is  to  the  intereft  of  100  dollars,  at  the 
fame  rate  and  time  ;  fo  is  the  given  fum,  to  the  Re 
bate  :  And  from  the  given  fum  fubtrad  the  Rebate, 
and  the  remainder  will  be  the  prefent  payment. 

EXAMPLES. 

A,  hath  100  dollars  due  to  him,  to  be  paid  at  the 
end  of  2  years  j  but  his  debtor  agrees  to  make  pre 
fent  payment,  provided  A  will  make  a  Rebate  at  6 
dollars  per  cent  per  annum  ;  Required  the  Rebate. 


OPERATION'. 


OPERATION. 


100 
6=  ratio, 

600 


12,00 

2  —  112 

12 


IJ2)l200(l0.7l 

112  [rebate 

80.0 

784 

1  60 

XI2 


Required  the  Rebate  of  720  dollars,  for  2.1  years, 
at  6  per  cent  per  annum. 


OPERATION. 


OPERATION. 
^100+15=115 : 15: 1720 

'5 

3600 
720 


115)10800(93.91  =  re- 
1035        \iate  req. 

450 

15.00  345 

105.0 
I03S 

150 
"5 

35 

Find  what  fum  ought  to  be  paid  down  for  a  debt 
of  1000  dollars  due  3^  years  hence,  difcQUHting  at  5 
per  cent  per  annum. 

PERSON* 


xoo 
5 

2)500 
3 

1500 
250 


17.50 


Aa 


'( J86      ) 

100+17 .50=117. 50  :  17.50  ::  rooo 

IOOO 


117,50)1750000(148.93 

11750 
57500 

47000 

105000 
94000 


I  IOOO.O 

105750 

42500 
3525° 


7250 
1000 — 148.93=851  doL  7  cts.  the  anjwer. 

Suppofe  I  have  a  legacy  due  to  me  of  4000  dollars, 
whereof  800  dollars  is  to  be  p'aid  in  8  months,    and. 
the  reft  at  the  end  of  16  months  :  How  much  ought 
I  to  receive  for  prefent  payment,  allowing  6  per  cent, 
&c.  difcount  ?  Anjwer.     3732  doL  21  cts. 

A  owes  B  15000  dollars,  one  half  of  which  is  to  be 
paid  in  4  months,  and  the  reft  at  the  end  of  8  months  : 
What  ought  B  to  receive  Fn  p!£fent  payment,  allow 
ing  6  per  cent  difcount  ?  <dnf*  14564  doL  58  cts* 


CHAP 


(      i*7.      ) 

C  H  A  P.     VI. 


of    P  ATMENTS 
<?be    COMMON    WAT. 

EQUATION  of  payments,  is  when  feveral  fums  of 
money  are  due  at  different  times,  to  find  a  certain 
time  when  the  whole  may  be  paid  without  lofs  to 
either  party 

RULE. 

MULTIPLY  each  payment  with  its  refpedtive  time, 
anddiyi.de  the.  fum  of  the  products  by  the  fum  of  the 
payments  5  the  quotient  refill  ting,'  will  be  the  time 
required. 

THIS  rule  will  give  the  equated  time  near  .enough 
for  common  practice  in  matters  of  this  nature  ;  but 
not  accurately  true,  becaufe  the  rule  is  founded  on  a 
fuppofition  that  the  fum  of  the  interefts  of  the  debts 
due  before  the  equated  time,  computed  from  the 
times  they  become  due  to  that  time,  is  equal  to  the 
fum  of  the  intereft  of  the  debts,  payable  after  the 
equated  time,  computed  from  that  time  to  their  re- 
fpective  terms  of  payment  ;  that  is,  the  gain  made  by 
the  debtor's  keeping-  thofe  debts  which  become  due 
before  the  equated  time,  until  that  time,  is  equal  to 
the  lofs  fuftained  by  paying  thofe  debts  at  the  equated 
time,  which  are  not  due  till  afterwards  ;  but  it  ismani- 
feft,  that  the  gain  made  by  keeping  a  debt  any  time 
after  it  is  due,  is  equal  to  the  intereft  of  that  debt 
for  that  time;  but  the  lofs  fuftained  by  paying  a  debt 
any  time  before  it  becomes  due,  is  plainly  no  more 
than  the  rebate  of  the  debt  for  that  time  -,  and  fmce 
the  rebate  is  always  lefs  than  the  intereft  of  the  fame 

fum, 


fum,  it  follows  that  the  fupppfition  is  not  true,  andr 
consequently  the  rule  falfe. 

EXAMPLES  in  equation  of  payments  the  common 
way. 

A  owes  B  100  dollars,  whereof  50  dollars  is  to  be 
paid  at  the  end  of  4  months  and  the  reft  at  the  end 
of  8  months  :  Required  the  time  when  the  whole 
may  be  paid  without  lofs  to  either  party. 

OPERATION. 

Fir  ft  y  50X4=200  thejirft  payment  with  its  time  ; 

Secondlyy  50X8=1400  thejecond  payment  with  its 
time : 

fberiy  400  4-  200—600  thefum  of  the  produfts. 

Andy  504-50=1100  thefum  of  the  payments  : 

Confequently,  6oo-i-ioo=6  months,  the  time  re 
quired. 

W  owes  X  865  dollars,  whereof  50  dollars  is  to 
be  paid  prefent,  195  dollars  to  be  paid  in  8  months, 
and  the  reft  at  the  end  of  12  months  :  Required  the 
equated  time  to  pay  the  whole . 

OPERATION. 

50X1—50  the  fir  ft  payment  with  its  time  :. 
195X8—1560  the  fecond payment  with  its  time  : 
62oX  1211:7440  the  laft payment  with  its  time  : 
And  5O-(- 1 560 +7440^:9050  the  Jum  of  the  pro- 

dutts. 

Confequentfyy  ?|4T^10  won$s  13  ll\days  the  tim? 

required. 

P  owes  a  debt  to  be  paid  at  5  feveral  payments, 
in  the  following  manner,  to  wit,  4  in  4  months,  ^  at 
8  months,  4  at  12  months,  |  at  16  months,  and  4  at 

20 


20  months  :  Required  the  equated  time  to  pay  the. 
whole. 

OPERATION. 

Suppofe  the  debt  =25  dollars,  one  ffth  of  which 
is  $  dollars,  then  5  X4+  5X^+5X1  2+5  Xi6  +  5X 
10=20+40+60+80+10011:300.  Therefore,  3|4z:- 
12  months,  the  time  required. 

IN  the  iblution  of  the  above  queftion,  we  madeufe 
of  25  dollars  to  reprefent  the  whole  debt  ^  but  any 
other  number  would  have  equally  fucceeded,  as  may 
be  thus  analytically  demonftrated. 

Let  x~anyfum  whatever,  to  be.  paid  in  manner  as 
above. 

jut  *  i  v*  v»  v* 

xis~>    and  7X4+7X8  -f-^X  12+  T  X  16 


,  ,  ,  -       ^ 

—  4-  —  4-  -  •+  -  =•  —  =fumof 

tleprodufts  of  the  fever  al  payments  with  their  refpeft- 

ive  times  :  Therefore,  --  r^=  -  —  —  ^-=  i  imonths 

5  5*       * 

tbefame  as  before.  <3   E  D 


CHAP.     VII, 

B  4  R  T  E  R. 

BARTER  is  the  exchanging  one  commodity 
for  another  in  fuch  a  manner,  that  the  parties 
bartering,  may  neither  of  them  fuftain  lofs.     Thus, 
fuppofe  A  hath  5olb.  of  ginger,  at  30  cents  perlb. 
and  would  Barter  with  B  for  pepper  at  70  cents  per 

Ib. 


Ib.  What  quantity  of  pepper  mud  B  give  A  for  his 
50  Ib.  .of  ginger  ? 

IN  the  Iblucion  of  this  queftion,  and  all  others  of 
the  like  nature,  you  muft  firft  find  the  value  of  the 
given  quantity  at  the  given  price,  and  then  find  how 
much  of  the  quantity  fought  at  its  price,  will  amount 
to  the  value  of  the  given  quantity,  and  the  refult  will 
be  the  a-nfwer  to  the  queftion, 

Thus,  in  the  above  queftion,  the  quantity  given,  is 
5clb.  of  ginger,  at  30  cents  per  Ib.  and  the  quantity 
fought  is  pepper,  at  70  cents  per  Ib.  Therefore,  as 
70  cts.  :  lib.  ::  15  dol.  (the  price  of  the  ginger)  : 
214  Ib.  the  quantity  of  pepper  required.  Oonfe- 
quently  in  Barter,  the  method  of  operation  is  the 
fame  as  i:i  the  rule  of  three  direcl. 

EXAMPLES. 

Required  the  quantity  of  flax,  at  §  cents  per  Ib. 
thatmuft.be  given  in  Barter,  for  12  Ib.  of  indigo,  at 
2  dol.  50  cts.  per  Ib. 

OPERATION. 

.Ib.  doL  cts.    Ib.   dol, 
Firft,  as  i  :  2  50  :;  12  :  30,  the  value  of  the  indi* 

go 
ers.  Ib.    dol.     Ib. 

?hen>  8  :  i  ::  30  ".  375,  the  anfwer. 

A  hath  rum  at  70  cents  per  gallon  ready  money, 
but  in  Barter  he  muft  have  So  cents  ;  B  hath  raifins 
at  1 2  cents  per  Ib.  ready  money:  How  many  Ib.  of 
raiiins  mufl  A  have  for  60  gallons  of  rum. 

Here  you  mufl  fir  ft  find  what  B's  raifins  ought  to  le 
fer  Ib.  in  Barter •,  which  muft  be  as  much  more  in  fro- 
fortion>  as  Ays  price  in  ready  money >  is  to  his  price  in 

Barter  i 


(      '9*      ) 

Barter ;  which  to  obtain,  Jay  as  70  cis.  '.  80  ct?.  ::  ia 
c/j. :  13.71  cts. ~ price  of  B's  raijins  $er  Ib.  in  Barter ; 
/#£#  proceed  as  before  diretled>  and  the  quantity  cfrai- 
fins  that  B  muft. give  A  will  be  found=3$o.iM. 

How  much  wheat  at  91 J-  cts.  per  bufhei,  mud  be 
given  for  8  ewt.  of  fugar  at  8^-  cts.  per  Ib.  ? 

Avfwer.     8i~  bujhels. 

A  hath  rum  at  70  cents  per  gallon  ready  money, 
bin  in  Barter  he  mud  have  84  cents  ;  B  hath  corn  at 
50  cents  per  bufhei  ready  money  :  How  muchmuit 
B  have  per  bufhei  in  Barter  for  his  corn  \  alfo,  how 
many  bufhei  of  corn  B  muft  give  A  for  a  hogfhead 
of  rum  containing  120  gallons  ? 

Anjwer.  M^n^ft  have  57^-  cts.  per  bitfoel  in  Barter, 
and  muft  give  A  168  kujbel  of  corn  for  the  12.0  gallons 
of  rum. 

D  hath  12  cwt.  of  fugar,  which  he  will  fell  to  H 
for  8  dollars  33-]-  cents  per  cwt.  ready  money,  but  in 
Barter  he  muft  have  8-i  cents  per  Ib.  H  hath  a 
horfe  which  he  would  fell  for  90  dollars  ready  mon 
ey,  but  in  Barter  he  muft  have  20  per  cent  advance: 
They  Barter,  D  takes  the  horfe,  and  H  the  fugar  : 
Query  which  is  in  debt,  and  how  much  ? 

Anjwer.  H  is  in  debt  3  doL-yj^  cts.  ready  money. 


CHAP.     VIII. 

LOSS   and    GAIN. 

LOSS  and  gain  is  a  rule  by  which  merchants' 
are  inftrufted  how  to  raife  or  fall   in  the  prices 
of  their  goods,  fo  as  to  gain  or  loofe  fo  much  per  Ib. 
bag,  or  barrel,  &c. 

The 


THE  operations  are  performed  by  the  rule  of  three 
direct. 

EXAMPLES. 

Suppofe  I  buy  cheefe  at  6  dollars  per  loolb.  and 
fell  it  again  at  8  cents  per  Ib.  What  do  I  gain  in  buy 
ing  and  felling  6oolb.  ? 

Here  you  muft  firft  find  what  6oolb.  comes  to, 
at  6  dollars  per  loolb.  and  6oolb.  at  8  cents  per  Ib. 
then  fubtra6t  one  fum  from  the  other,  and  the  refult 
will  be  the  anfwer. 

OPERATION. 

Firft,  6X6—36  doL  the  value  of^6volb.  at  6  doL 
per  ioolb. 

Then,  600X8  cts.~  48  doL  the  fries  of  6oolb.  at 
8  cts.  per  Ib. 

And,  4$— 36—11  dol.  the  anfwer. 

When  butter  coft  7  dollars  per  firkin  of  561b.  To 
find  how  it  muft  be  fold  per  Ib.  to  gain  25  per  cent. 

OPERATION. 

<* 

As  $6lb.  :  7  dol. ::  lib. :  iz~cts.  the  price  that  tfa 
butter  coft  per  Ib. 

Asioolb.:  ii^tts.  ::  1004-25=125  :  15.625  cts. 
the  anfwer* 

When  tea  coft  75  cts.  per  Ib.  To  find  how  it  muft 
be  fold  per  Ib.  to  gain  25  per  cent. 

OPERATION. 

As  100  :75  ::  100+25=125:  93.75  cts.  the  an- 

Cwer* 

At 


C      *93      ) 

At  I2~  cts.  profit  in  a  dollar  :  How  much  per  cent? 

As  i  dbL  :  12.5  cts.  ::  100  dol.  ;  12- per  cent  the 
'anfwer. 

Bought  rum  at  50  cents  per  gallon,  and  paid  i'm- 
poft,  at  $  cents  per  gallon,  and  afterwards  fold  it  at 
53  cents  per  gallon :  What  do  I  loofe  in  laying  out 
€00  dollars.  Anfwer.  86  dol.  21  cts« 

If  I  buy  tallow  at  ii~  cents  per  Ib.  and  give  2%. 
cents  per  Ib.  to  a  chandler  to  make  it  into  candles, 
and  i4oz.  of  tallow  make  a  dozen  of  candles,  which 
I  fell  at  1944  cents  per  dozen  :  What  do  I  gain  in 
•buying  and  felling  iSolb.  of  tallow. 

Arifwer.     iQ.dcl.$Qcts. 


CHAP.       IX. 

FFLLOWSHIP, 

FELLOWSHIP  is  a  rule,  when  feveral  per- 
fons  as  merchants,  &c.  trade  in  company  with 
a  joint  flock,  to  afcertain  each  man's  proportional 
part  of  the  gain  or  lofs,  which  arifes>from  the  em 
ployment  of  the  joint  (lock,  according  to  the  quan 
tity  of  goods,  Hum  of  money,  &c.  each  man  puts  in 
to  the  faid  ftock  j  which  admits  of  a  two-fold  con- 
fideration. 

SECT,      L 
FELLOWSHIP     SING  LE. 

SINGLE  Fellowfriip  is  when  all  the  feveral  (locks 
are  employed  in  the  common  (lock,  an  equal  term 
of  time.  Therefore,  fince  the  times  of  the  feveral 

B  b  flocks 


(      1.94      ) 

flocks  employed  in  the  joint  flock,  are  all  equal  ;  it 
follows,  that  each  partner's  Ihare  of  the  gain  or  lofs, 
is  as  his  Ihare  of  that  Hock  :  Wherefore  it  is  mani- 
fed  ;  if  I  put  in-^  of  the  whole  Hock,  I  ought  to 
have  -^  of  the  whole  gain,  or  fuffer  ^  of  the  whole 
lofs  :  Hence  ws  have  the  following 

RULE. 

MULTIPLY  each  partner's  part  of  the  joint  flock, 
with  the  vdiole  gain  or  lofs,  and  divide  the  feveral 
products  by  the  whole  flock,  and  the  quotients  re- 
fulting  will  be  the  anfwer  to  the  queftion.  Or,  as  the 
whole  flock  is  to  the  whole  gain  or  lofs  j  fo  is  each 
man's  particular  part  of  that  flock,  to  his  particular 
part  of  the  gain  or  lofs* 

EXAMPLES. 

Two  partners,  A  and  B,  conftitute  a  joint,  flock  of 
300  dollars,  whereof  A  put  in  200  dollars,  and  B 
100  dollars,  and  they  trade  and  gain  150  dollars  : 
Required  each  man's  part  of  the  gain. 


OPERATION. 


150 
100 


3)00)300)00  3)oo)  1 50)00 

i  oo—  A's  gain.  50=  B*s  gain. 

Or, 

As  300  :  150  ::  200  :  150X200^300=100  doL 
A's  fart  of  the  gain. 


As  300:  150::  ioo  :  1 50 xi 00^-300—50 dol.  B's 
fart  of  the  gain  : 

Or, 


C      *9$      ) 

Of,  150—300:3.5.  the  ratio  of  the  fir  ft  term  to  the 
Jecond  : 

Therefore,  iQQX.$~ioo  A's  part,  and  ioox«5  = 
50  B's  part  as  before.  (Vid.  Chap,  u.) 

Three  merchants,  A,  B,  andC,  make  a  joint  ftock 
of  2000  dollars,  whereof  A  put  in  1500  dollars,  B  800 
dollars,  and  G  700  dollars  \  and  by  trading  gain 
400  dollars  :  Required  each  man's  part  of  the  gain  ? 

OPERATION. 

Fir  ft,  40O-r  aooo—.a  the  ratio  of  the  fir  ft  term  to 
thefetond. 

f  500X.2—IOQ  dol  A's  "I 
'therefore^  <   8oox-2—  160  —  —B's  >  gain. 
(.  yooX.  2-140  -  C's]  ' 


Four  merchants  enter  into  partnerfhip,  and  confti- 
tute  a  joint  (lock  of  60000  dollars,  whereof  A  put  in 
15000  dollars  24  cents,  B  20000  dollars  76  cents, 
C  21000  dollars,  and  D  3999  dollars,  and  in  trade 
they  gain  24000  dollars  :  Required  each  partner's 
(hare  of  the  gain  ? 

OPERATION. 

Firft,  24000-7-60000^.4  the  ratio  of  gain  :  There 
fore,  1  5000.  24X.  4=6000  dol.  9.6  cts.  Ays  part  of 
the  gain  -,  and  20000.7  6  X'4—  8000  dol.  30.4  cts.  B's 
part  of  the  gain  ;  alfoy  2  ioooX  -4=8400  dol.  C's 
part  ;  laftly,  3999X«4=i599  del.  60  cts.  D's  part. 

Six  farmers,  A,  B,  C,  D,  E,  and  F,  hired  a  farm 
for  300  dollars  ;  A  paid  20  dollars,  B  30,  C  40,  D 
60,  £  80,  and  F  70  dollars  ;  and  they  gained  60 
dollars  :  What  is  each  man's  part  of  the  gain  ? 

Answer. 


c   *?g  ) 

4  */•  5V  6,  C'jf  «.,  P>  xa£E'j  56, 
&nt?F's  14  <£?/. 

SECT-      JI> 
CO  MPOUND    FELL  O  WSHIP. 

TnEonly  difference  between  Fcllowfhip  fingle  and 
compound,  is,  that  in  the  latter  regard  muft  be  had 
to  the  time  each  partner's  fjtock  continues  in  com 
pany  5  whereas  in  fingle  Fellowfhip  the  times  of  con 
tinuance  are  all  fuppofed  equal*  and  when  the  times 
are  equal,  the  {hares  of  gain  or  lofs,  are  as  their 
flocks,  as  we  have  before  fnewn  :  Therefore  when 
the  flocks,  are  equal,  the  fhares  muft  be  as  the  times. 
Confequently,  when  neither  the  ftocks  nor  times  are 
equal,  the  fhares  muft  be  as  their  products  \  which 
affords  the  following 

RULE. 

1.  MULTIPLY  each  man's   flock  with  the  time  it 
is  employed,  and  find  the  fum  of  all  the  products. 

2.  As  the  fum  of  the  products  thus  found,   is  to 
the  whole  gain  or  lofs  ;  fo  is  the   product  of  each 
inan's  flock  with  its  time^   to  its  proportional  pajt 
of  the  gain 'or  lofs. 

Or, 

FIND  the  ratio  between  the  two  £rft  terms,  and 
proceed  as  in  the  laft  rule, 

EXAMPLES. 

Two  men,  A  and  B,  made  a  joint  flock  of  600 
dollars,  whereof  A  put  in  200  dollars  for  2  months', 
B  put  in  400  dollars  for  4  months  -,  at  the  expir 
ation 


ation  of  which,  they  find  they  have  loft  200  dollars* 
Required  each  man's  part  of  the  lofs  ?< 

OPERATION. 

Firft,  200x2—400:3  A's  ftock  with  its  time  : 
Andy  400X4=1600  —  B's  ftock  with  its  time  : 
Then,  4004-1600^:2000  the  fum  of  the  prcdufts  of 
each  man's  ftocky  with  its  time :  'Therefore)   as  2000  : 

200  ::  400  :  2oox400-J-2:ooo~4O  del.  A's  fart  of  the 


lofs;  and  as  2000  :  200  ::  1600  :  200X1600-7-2000 
—  160  dol.  B  'j  fart  of  the  lofs. 

Or,  2oo-r-2ooo  ~  .1  the  ratio  cf  lofs  9  then, 
40oX.i~4O  A' s  party  and  i6ooX«i~i6o  B's  part, 
the  fame  as  before. 

Three  merchants  made  a  joint  flock  of  8000  dol 
lars  in  the  following  manner^  viz.  A  put  in  1200  dol 
lars  for  3  years,  B  2000  dollars  for  7  years,  and  C 
4800  dollars  for  8  years  •>  and  at  the  end  thereof, 
they  find  they  have  gained  6720  dollars:  Required 
e&cfx  man's  part  of  the  gain  ? 

OPERATION. 

Firft,  i2OoX3~36oozi^'j  ftock  with  its  time  :•. 
And,  2000X7 ~i4000rn$\f  j#0f&  with  its  time", 
AlfOy  4800X8—38400— Cs  ftock  with  its  time  : 
Then,  3600+ 14000  +  38400—56000  the  fum  cf  the 

yrodufts  : 

Andy  67  20  -r-  5600011:.  12  the  ratio  of  gain  : 
Therefore,   3600  X- 12:1:432   dol.    A* s  part   of  the 

gain-,   and  I4ooox-I2=ri68o  doL    B's^party  Alfa^ 

3S400X-12— 4608  doL  Csfart. 

Two- 


Two  merchants,  A  and  B,  made  a  joint  ftock  ;  A 
put  in  at  firft,  300  dollars  for  7  months,  and  4 
months  after  put  in  500  dollars  more  :  B  put  in  at 
firft,  700  dollars,  and  3  months  after  put  in  200  dol 
lars  more.  Now  at  the  end  of  7  months,  they  make 
a  fettlement  of  their  accounts,  and  find  they  have 
gained  1860  dollars  :  Required  each  man's  part  of 
the  gain,  according  to  his  (lock  and  time  ? 

F*r ft  *  300x4—1200  the  produft  of  A' s  fir  ft  flock 


with  its  time.)  and  3004-500X3—800x3=2400  the 
product  of  As  increafed  ftock,  with  the  remainder  of  the 
time  :  Therefore,  12004-2400—3600  the  produtt  of 
Ay s  flock  with  the  whole  time,  according  to  the  queftion. 

Secondly,  700X3=2100  the  produft  ofB's firft  fleck 


with  its  time,  and  7004-200X4—900X4—3600  the 
frodyfaof  B's  augmented  ftock  y   with  the  remainder  of 
the  time  :  therefore,   21004-3600=5700  the  product 
cf  B's  whole  ftock,  with  the  whole  time,  and  36004- 
5700^9300  thejum  of  the  produtls. 

Hence  ^  1860—03001^.2  the  ratio  of  gain  :  there 
fore,  36ooX.2i=72o  dol^A's  part  of  the  gain,  and 

of  the  gain. 


Four  merchants,  A,  B,  C  and  D,  enter  into  part,- 
nerfhip  for  12  months:  A  put  into  the  common 
ftock  at  firft,  300  dollars,  B  400,  C  500,  and  D  800 
dollars,  and  at  the  end  of  four  months,  A  took  out 
200  dollars,  and  3  months  after  that,  he  put  in  100 
dollars  more  -,  Bat  the  end  of  2  months  took  out 
200  dollars,  and  2  months  after  that,  put  in  200  dol 
lars  more  :  C  at  the  end  of  6.  months,  took  out  300 
dollars,  and  two  months  after  that,  put  in  200  dol 
lars  more  :  D  at  the  end  of  8  months,  took  out  400 
dollars,  and  2  months  after  that,  put  in  200  dollars 

more  : 


*(      199      ) 

more  :  Now  at  the  end  of  12  months;  they  find  they 
have  gained  406  dollars  :  Required  each  man's  part 
of  the  gain  ? 

OPERATION. 
Firfl,  300X4=1200  the  produft  of  A'sjirft  flock 


with  its  time,  and  300 — 200X3  —  100X3  —  300  the 
frodutt  of  A's   remaining  fleck  for  3  months  after  the 


taking  out  of  the  200  doL  Again,  ioo-j-iooX5  = 
200X5=1000 /£<?  produft  of  A's /lock  with  the  re 
mainder  of  the  time  according  to  the  queflion  ;  then, 


1200+300+1000=2500  the  produft  of  A's  flock  for 
the  whole  time. 

Secondly,  to  obtain  the  produfit  of  B's  flock  with  its 
time,  proceed  as  before  :    Thus  400X2  =  800  ;  then, 


400—200X2=200X231400;  and   200  +  200X8  — 


400X8  =  3200.  Hence,  8004-400  +  3200=4400  the 

froduff  of  E's flock  with  its  time. 


Thirdly,  500X6=30005  then  500 — 300X2—200 


and  2004-200x4=400X4=1600  ; 
wherefore  3000+4004- 1600=5000  the  $  rodutt  of  Cs 
ftock  with  its  time. 


Fourthly,    800X8=6400  ;     then  800— 400X2:= 


400x^=800;  and  400+200X2=1600X2—1200; 
therefore  6400  +  8004-1200=8400  the  frodutl  of  D's 
Jlock  with  its  time. 

Confequently, 


(      ado     ) 

-Confiquentl?,  2500 +4400  +  5000+ 8400 ~  2030*0 
the  fum  $f  all  the  prcrdufis  according  to  the  queftion. 

Therefore,  406-7-203002^.02  the  ratio  of  gain  y  and 
250oX-02rr5o  doL  A' s  part  of  the  gain  \  aljo,  440OX 
.O2rr88  doL  B'f  part  ,  likewifa  5000X^02—100 
4ol.  C'spart  ;  laflly,  8400X^02=168  dol.  D'spart. 


CHAP     X. 

CO  MP  0  UND      PROPORTION. 

COMPOUND  Proportion,    is  ufed   in  the 
folution  of  queftions  that  require  feveral  oper 
ations  in  fimple  proportion,  \vtiether  diredt  or  reci 
procal. 

FOR  inftance  :  Suppoie  a  footman  performs  a 
journey  of  240  miles  in  8  days,  when  the  days  arc 
1 6  hours  long  :  In  what  time  would  he  perform  a 
journey  of  540  miles,  when  the  days  are  but  12  hours 
long.  This  queftion  refolved  by  fimple  proportion 
is  thusj 

m.     d.       m.    54oX8_ 
/&  240  :  8  : :  540  :    24Q     —18  days. 

THAT  is  it  would  require  18  days  to  perform  a 
journey  of  540  miles,  when  the  days  are  18  hours 
long  i  but  it  is  required  to  know  how  many  days  it 
will  take  to  perform  the  laid  journey  of  540  miles 
when  the  days  are  but  1 2  hours  long  5  which  is  thus  : 


As  i6£.  1540X8 -7-240  ( 1 8d.)::  12:  540X8X12-7- 


240  x  1 2^:24  day s 1 1)'  inverfe  proportion, 

Now 


Now  from  the  laft  analogy,  is  deduced  the  fol 
lowing  rule,  for  ftating  and  working  all  queftions  in 
compound  proportion,  at  one  operation. 

RULE. 

i.  PLACE  that  term  which  is  of  the  fame  name  of 
the  term  lought,  fo  that  it  may.ftand  in  the  middle 
place :  d. 

Thus,   •< 

2*  WRITE  the  remaining  terms  of  fuppofition,  one 
above  the  other  in  the  firft  places,  and  the  terms  of 
demand-  in  like  manner  in  the  third  places,  fo  that 
the  firft  and  third  terms  in  each  row,  may  be  of  the 
fame  name  and  denomination  : 

m.     d.      m. 
r  240 :  8  : :  540 
Thus,  4     h.  h, 

L  16: ::i2 

3.  HAVING  thus  ftated  your  queftion,  find  your 
divifOr  by  comparing  the  terms  in  each  row  :  Thus 
if  the  firft  term  gives  thefecond,  does  the  third  term 
require  more  or  lefs  ?   If  more,  diftinguifli  the  lefs 
extreme  with  a  point  over  it  •>  but  if  the  third  term 
require  lefs,  point  the  greater  extreme  : 

.  m.  d.      m. 
f  240  :  8  : :  540 
Thus,  \      h.  .h. 

I   16: ::  12 

4.  MULTIPLY  together  the  terms  which  are  point 
ed  for  a  divifor,  and  the  remaining  terms  .for  a  divi 
dend,  and  the  quotient  refulting  will  be  the  anfwcr : 

Thus,  540X8X16-- 240X1 2—24  days  as  before, 
C  c  EXAMPLES, 


(      202      ) 


EXAMPLES. 

If  12  bufhels  of  corn  are  fufficient  for  a  family  of 
9  perfons  12  months  :  How  many  bufhels  will  be 
fufficient  for  a  family  of  16  perfons,  20  months  ? 

OPERATION. 


Here  lujhels  are  fought  •>  therefore  the  queftion  ftated 
willjland 

.per.  b.  per. 
r  9 :  12::  1.6 
Thus,  <    .  m.  m. 

(.  12  : ::20 

f hen  Jay,  if  9  perfons  eat  1 2  lujhels  in  1 2  months, 
1 6  perfons  will  edt  more  5  therefore  point  the  lejs  ex* 
ireme,  which  is  9.  Again >  Jay y  if  12  months  require 
1 2  bu/hels  for  9  perfons,  20  months  will  require  more  \ 
therefore  point  the  lefs  extreme,  which  is  12. 


Therefore,  12X20X16-- 12X9—3840-7- 108=35!. 
lujhels ,  the  quantity  of  for  n  required. 

Note.  If  the  fame  quantity  is  found  loth  in  the  divifor 
and  dividend,  it  may  be  expunged  from  loth  : 


/.>  the  above  exprefflon,  1  2X  20  x  *  6-f-  1  2X9,  /fij 
1  2  wtfj  be  flruck  out  of  the  divifor  and  dividend  ;  thus, 


2oX  1  6  -^9^:3  5-1  the  fame  as  lefore. 

If  15  dollars  be  the  hire  of  8  men  5  days  :  What 
time  will  40  dollars  hire  20  men  ? 


OEERATION. 


(      2°3      ) 

OPERATION. 

.do  1.  d.     dot. 
IS  :  51:40 
m.  m  m. 

8  : ::20 


15X20=1600^300=51^^, 
the  time  required. 

If  200  dollars  in  2  years,  gain  15  dollars  :  What 
will  150  dollars  gain  in  half  a  year  ? 


thus,    15x150X26-7-200X104=  i^doL   the 
anfwer. 

If  i50olb.  of  bread  ferve4Oo  men  14  days  :  How 
many  pounds  of  bread  will  ferve  140  men  9  days  ? 


Thus,  1  500x140X9-7-  400  xi4==337#»  %oz.  the 
anfwer. 

If  12  Clerks  will  write  72  fheets  of  paper  in  3 

*days  :  How  many  Clerks  will  write  140  Iheets  in  8 

days  ? 


Anjwer*   ^XjXHO-r-y^X8  —  8^  Clerks. 
If  5000  bricks  are  fufficient  to  make  a  wall  4  feet 
high  and  5  feet  long  :  How  many  bricks  of  the 
fame  fize  will  make  7  feet  of  wall  2  feet  high  ? 

Anfwer.  3500, 


CHAP, 


204 


CHAP.       XL 

CONJOINED    PROPORTION. 

CONJOINED  Proportion  is  when,  in  a  rank  of 
numbers,  the  firfl  term  is  compared  with  the  fecond, 
and  the  fecond  term  being  increafed  or  diminiihed, 
is  compared  with  the  third,  and  fo  on  $  from  thence 
to  determine  the  equality  of  any  of  the  terms  :  Thus, 
ifj^~4^,  and  8£~i2r,  then  will  3a=6c  -,  becaufe, 
as  4#  :  30  :  :  8£  :  60—  izc,  or  ^a^Jbc  as  before.  A- 
gain,  if  240—  3  2#,  48^^:30^  and  ioc—$d,  then  will 
becauie,  as  32^  :  24^  : 


=36^1^  yx,  and 


a4«=20<r.   Again    as   jor :  24^X48^-7-32^  :: 


=i2^~9,  or  24^— 
Hence  from  the  foregoing  analogy  we  have  the  fol 


lowing 


RULE. 


1.  BEGIN  with  that  term  whofe  equality  with  any 
Other  term  is  required,  which  call  A  and  write  out 
all  the  terms  up  to  the  one  B,  by  which  the  afore- 
faid  term  is  to  be  compared. 

2.  MULTIPLY  all  the  alternate  numbers  together, 
beginning  with  the  firfl,  for  a  dividend,  and  all  the 
remaining  ones  together  for  a  divifbr. 

3.  Divide,  and  the  quotient  will  be  the  anfwer, 

EXAMPLES. 


EXAMPLES. 

If  G  in  48  days  can  produce  a  certain  effect,  which 
will  require  H  64  days  to  perform  j  H  can  pro 
duce  an  effect  in  80  days,  which  will  take  JL  50  days 
to  perform  :  Which  is  the  mod  profitable  to  hire, 
G  or  L,  and  what  is  the  difference  ? 

OPERATION. 

Fir/},  48,  64,  80,  are  the  numbers  written  out  ac 
cording  to  the  rule : 

<Tben,  48x80--- 64=60— 50  days' of  L,  that  is,  60 
days  of  G  are  equal  to  50  days  of  L ;  and  therefore  if 
is  the  moft  profitable  to  hire  L,  to  wit,  in  the  proportion 
of  60  to  50,  or  as  6  to  5. 

IfD  in  24  days  can  do  as  much  as  E  can  in  32 
days,  E  can  do  as  much  in  48  days,  as  F  can  in  30 
days,  and  F  can  do  as  much  in  10  days,  as  G  can  in 
9  days  :  Which  is  the  moil  profitable  to  hire,  D,  F, 
orG  ? 

OPERATION. 
Fir  ft,  find  which  is  the  moft  profitable  to  hire,  D  orF: 

Thus,  24X48-4-32=36=30  days  of  F,  that  is,  36 
days  cfD  are  equal  to  30  days  of  F  -,  and  therefore  F 
is  more  profitable  to  hire  than  D, 

Again,  24x48X10-7-32X30=12:1:9  days  of  G  ; 
that  is,  1 2  days  of  D  are  equal  to  9  days  of  G>  and 
therefore  G  is  more  profitable  to  hire  than  D  ;  andfince 
F  is  more  profitable  to  hire  than  Dy  and  G  more  profit 
able  than  F  j  if  fellows,  that  G  is  the  moft  profitable  to 
hire  of  the  three.  CH  A  P. 


(        206        ) 

CHAP.     XII. 

ALLEGATION. 

BY  Allegation  we  are  taught  how  to  mix  quan 
tities  of  different  quality,  fo  that  any  quantity 
colleftively  taken,  may  be  of  a  mean  or  middle 
quality  ;  that  is,  it  fliews  us  the  value  of  any  part  of 
a  competition,  made  of  things  all  of  a  different 
quality. 

WE  fhall  confider  Allegation,  under  the  two  fol 
lowing  general  heads,  viz.  Allegation  Medial,  and 
Allegation  Alternate. 

SECT.       I. 

ALLEGATION   MEDIAL. 

THIS  is  when  any  number  of  things  are  given,  and 
the  price  of  each  :  To  find  the  price  of  any  quantity 
of  a  mixture  compounded  of  the  whole. 

RULE. 

1.  MULTIPLY  each  quantity  with  its   price,   and 
find  the  fum  of  all  the  proslufts. 

2.  DIVIDE  the  fum  of  the  produces  by  the  fum  of 
all  the  quantities,  and  the  quotient  refulting  will  be 
the  mean  price  required. 

EXAMPLES. 

A  man  is  minded  to  mix  20  buftiels  of  wheat,at  100 
cents  per  bufhel,  with  10  bufhels  of  rye,  at  50  cents 
per  bufhel :  Required  the  price  of  a  bufhel  of  this 
mixture.  OPERATION. 


(      207      ) 


OPERATION. 

Firfl,  20X100=2000  cts.  —  price  of  all  the 
wheat,  and  10X50—500  cts.  —  frice  of  the  rye  •  then 
2000+500—2500  thefum  of  the  produces,  and  20+ 
10—30  thefum  of  the  quantities  :  Therefore,  2500-- 
30—83]-  cts.  the  price  tf  a  lujhel,  as  was  required. 

A  man  would  mix  27  bufhels  of  wheat,  at  75  cents 
per  bufhel,  with  40  bufhels  of  rye,  at  60  cents  per 
bufhel,  and  24  bufhels  of  oats,at  24  cents  per  bufhel  : 
Required  the  price  of  a  bufhel  of  this  mixture. 

OPERATION. 
9 

Firft)  27X75z::i885  cts.  —price  of  the  wheat,  and 
40X60—2400  cts.  —  price  of  the  rye,  alfo,  24X24— 
576  cts.  —  the  price  of  the  oats  ;  then  1 88 5-!- 2400-!- 
576—4861  thefum  of  the  products,  and  27+40  +  24 
—  91  thefum  of  the  quantities. 

Whence  4861  -cts.-r-yi^z price  of  a  bujhel,  as  was 
required. 

A  mahfter  would  mix  70  gallons  of  one  fort  of  beer, 
worth  12  cents  per  gallon,  with  20  gallons  of  another 
fort,  worth  24  cents  per  gallon,  and  20  gallons  of  a 
third  fort,  worth  22  cents  per  gallon  :  How  may  this 
mixture  be  fold  per  gallon  without  gain  or  lofs  ? 

Anfwer.     16  cts. 

Required  what  a  gallon  of  the  following  mixture 
is  worth,  viz.  60  gallons  of  malaga,  at  .5  dollars  per 
gallon,  40  gallons  at  .7  dollars  per  gallon,  and  12 
gallons  at  .3  dollars  per  gallon. 

Anfwer.     .55  dol. 

A  Goldlinith  melts  iSffi.  of  gold  bullion,  of  12 
carats  fine,  with  iofl>.  of  16  carats  fine,  and  20 ]fo.  of 

10 


(        203        ) 

10  carats  fine  :  How  many  carats  fine  is  a  pound  of 
this  mixture.  Anfwer.     1 2  carats. 

Note.  Goldjmiths  fuppoje  every  quantity  of  gold  to 
confifl  of  24  farts,  which  they  call  carats  ;  but 
gold  is  generally  mixed  with  fome  other  me  tats,  fitch 
as  copper^  Irafs,  &c.  which  is  called  alloy  ^  and 
the  quality  of  the  gold  is  eftimated  according  to 
the  quantity  of  alloy  in  it :  Thus  if  20  carats  of 
pure  gold,  and  4  of  alloy  are  mixed  together,  the 
gold  is  called  20  car  at  j  fine. 

SECT.      II. 

ALLEGATION    ALTERNATE. 
ALLEGATION  Alternate  confifts  of  3  cafes. 
CASE    I. 

When  the  prices  of  the  federal  quantities  to  be  mixed 
are  given y  to  find  what  number  ofcachjort  muft  be  ta 
ken,  to  ccmpofe  a  mixture  whcfe  mean  price  Jhall  be  as 
given  in  the  jueftion* 

RULE. 

1.  WRITE  all  the  particular  rate*  or  prices   di- 
reftly  under  each  other,  and  the  mean  price  on  the 
left  hand. 

r  i 

Thus,  mean  price,  4  <       particular  prices. 

I  5 

2.  COUPLE  or  connect  the  particular  prices  with 
lines,  fo  that  one  or  more  of  thole  greater  than  the 
mean  price,   may  be  coupled  with  one  or  more  of 
thofe  lefs.  Thus, 


Thus,  4 


Or  thus>4 


5-  L  5 

WRITE  the  difference  between  the  mean  price 
md  every  particular  price,  direclly  againft  the  one 
nth  which  it  is  coupled. 

fi— ni  fi 

I  A — \       <j  ^     .  I  ^ 

hus,  4 


4.  THE  difference  (landing  againft  each  particular 
price,  is  the  quantity  that  mutt  be  taken  of  that  kind  5 
and  where  two  or  more  differences  are  found  Handing 
againft  any  one  particular  price,  their  fum  is  the 

uantity. 

Amaltfterhas  the  following  forts  of  beer,viz.  at  12 
cents,  22  cents,  and  24  cents  per  gallon  :  Required 
the  quantity  of  each  fort  that  muft  be  taken  to  make 
a  composition  worth  20  cents  per  gallon. 


20 


y 

L  2 


OPERATION. 
2+4=6 


22 I 

24  — 


8 


Therefore,  there  muft  be  taken  6  gallons  at  1 2  cts.  8 
gallons  at  22  f/j.  tf«df  8  gallons  at  24  i:/j.  w/^/V^  w^ 
be  proved  by  Allegation  Medial. 

To  find  how  much  wheat  at  100  cents  per  bufliel, 
rye  at  75  cents,  corn  at  40  cents,  and  oats  at  30  cent* 
per  bufhel,  may  be  mixed  together,  fo  that  the  mix 
ture  may  be  fold  for  50  cents  per  bufhel,  without  gain 
or  lofs. 

D  d  OPERATION* 


(        210        ) 

OPERATION. 


anjwtr. 


A  merchant  has  coffee  worth  12,  15,  16,  and  ic 
cents  per  Ib.  and  would  make  a  mixture  worth  14 
cents  per  Ib.  What  quantity  of  each  fort  muft  be 
taken  ? 

OPERATION. 

Ib.       cts.  Ib.       cts. 

fi2— ^\    i    at  12  fi6 \2  at    16 

IISJ    a. 15  Ii5         14 15 

14  1  i6-n    4 16  (        ^,  12— J  2 12 

LTO_J    2 10  LIO — J      i 10 

Proceeding  in  this  manner,  by  varying  the  order  fif  link 
ing  the  particulars,  you  will  dif cover  five  more  anfwen 
to  this  queftion,  in  whole  numbers. 

How  thefe  kind  ofqueftions  can  admit  of  various 
anfwers,  is  eafy  to  conceive  -3  for  if  any  two  of  the 
particular  prices  make  a  balance  by  their  increment 
and  decrement,  in  refpecl  of  the  mean  price,  ther 
will  any  multiple  or  quotient  of  the  fame,  make  i 
balance  alfo  :  Therefore  all  numbers  which  are  ir 
the  fame  proportion,  equally  anfwer  the  queftion 
Confequently,  there  are  fome  queftions  which  wil 
admit  of  an  infinite  variety  of  anfwers  :  Hence  it  is 
that  thefe  queftions  are  fometimes  called  indetermin 
ate  or  unlimited  problems  -,  yet  by  an  analytical  pro 

cefs 


:efs,  we  can  difcover  all  the  poffible  anfwers  in  whole 
inumbers,  when  thofe  anfwers  are  limited  to  finite 
terms.  ( Fid.  Book  n,  Chap.  xxm.J 

CASE    II. 

When  tie  quantity  of  one  of  the  particulars  is  limit '- 
•d  or  given,  thence  to  proportion  all  the  others  in  the 
compofttion  by  it. 

RULE. 

1.  OBTAIN  the  difference  between  the  mean  price 
and  every  particular  price,  as  in  the  lad  rule. 

2.  As  the  difference  found  againft  thefimplewhofe 
quantity  is  givep,  is  to  the  quantity  itfelf  5  fo  is  each 
difference,  to  its  refpeftiye  quantity  of  the  compofi- 
tion. 

EXAMPLES. 

A  farmer  would  mix  12  bufhels  of  wheat  at  72 
cents  per  bufhel,  with  rye  at  48  cents,  corn  at  36 
cents,  and  barley  at  30  cents  per  bufhel,  fo  that  the 
whole  compofition  may  be  fold  for  38  cents  per 
bufhel :  Required  the  quantity  of  each  fort  that  mud 
be  taken. 

OPERATION. 


36- 


jo- 


8 

2 
IO 

34 


^  as  8  :  1 2 : :  2  :  3,  the  quantity  of  rye,  and>  as 
8  :  12 ::  10 :  15,  the  quantity  of  corn  j  alfo,  as  8  :  12 
; :  34  •  Si>  the  quantity  of  barley,. 

To 


To  find  how  many  gallons  of  frontenaic  at  81  < 
cents,  cFaret  at  60  cents,  and  port  at  51  cents  per1 
gallon,  mull  be  mixed  with  42  gallons  of  madeini 
at  90  cents  per  gallon,  fo  that  the  whole  compofition  | 
may  be  fold  for  72  cents  per  gallon,  without  profit! 
or  lofs. 

21 
12 

9 

18 


15* 


f ben,  42—21—2  ;  therefore,  12X2 —24,  the 
quantity  of  the  claret,  and  9x2:1:18,  the  quantity  of 
the  frwtinaic  >,  alfo,  18X2=36,  the  quantity  of  fort. 

A  tobacconift  would  mix  6  Ib.  of  tobacco  worth 
6  cents  per  Ib.  with  another  fort  at  M  cents,  and  a 
third  fort  at  12  cents :  What  quantity  muft  be  taken 
of  each  fort,  to  make  a  mixture  worth  10  cents  per 
Ib  ?  Anfwer,  8  Ib.  ofeachjort. 

CASE      III. 

When  the  whole  compofition  is  equal  to  a  given  quan 
tity  ;  that  is,  when  thefum  of  all  the  quantities  which 
make  up  the  compofition,  colletJively  taken,  amount  to 
the  given  quantity  :  'To  find  the  feveral  quantities  them- 
fefaes. 

RULE. 

1.  LINK  or  couple  the  feveral  particulars,  and  find 
their  differences,  as  in  thelaft  cafe. 

2.  As  the  fum  of  the  differences,  is  to  the  fum  of 
the  whole  compofition  or  given  quantity  ;  fo  is  each 
difference,  to  its  refjpeftive  quantity  of  the  compofi 
tion. 

f 

EXAMPLES. 


EXAMPLES. 

A  grocer  having  fugars  at  4  cents,  8  cents,  and  12 
cents  per  Ib.  would  make  a  compofition  of  240  Ib, 
worth  10  cents  per  Ib.  Required  the  quantity  of  each 
fort  that  muft  be  taken. 

OPERATION. 

Firft,   10  \    8— <|   I  2 

I  12— LJ  64-2=8 


12—  fum  efthe  differences. 

Theny  as  12  :  240  : :  t :  40,  andy  as  1 2:240  : :  2  :  40 ; 
alfoy  as  12  :  240 : :  8  :  160.  Therefore,  there  muft  be 
taken>  40  Ib.  at  4  cts.  40  Ib.  at  8  cts.  and  160  Ib.  at 
12  cts. 

A  merchant  would  mix  brandy  of  the  following 
prices,  viz.  at  60  cents,  72  cents,  and  84  cents  per 
gallon,  together  with  water  at  o  cents  per  gallon,  fo 
that  a  compofition  of  846  gallons,  may  be  fold  for 
48  cents  per  gallon,  without  gain  or  lofs  :  Required 
the  quantity  of  each  fort  that  muft  be  taken. 


OPERATION. 


Firft,  48 


Then,  <w.  216;  846:: 


48 
48 
48 
24-^X1+36=72 

216— fum  of  the  differences. 
'48  :  188  at  jicts. 

$„;  1 88  at  60  cts. 
48  :  1 88  at  84  cts. 
72  :  282  of  water.  In 


(        214        ) 

IN  this  cafe  might  beftarted,a  variety  of  very  curious 
queftions  about  the  fpecific  gravities  of  metals  3  but, 
as  they  would  require  the  knowledge  of  fom.e  things 
which  are  not  treated  of  in  this  volume,  we  defift. 


CHAP.     XIII. 

Of    POSITION,    or    the    GUESSING 
RUL  E. 

POSITION  is  a  method  of  folving  queftions, 
by  fuppofmg  numbers,  and  then  adding  them, 
fubtrading,  multiplying,  &c.  according  as  the  refult 
or  number  given  in  the  queftion  is  produced  by  ad 
dition,  fubtra6lion,  multiplication,  &c,  of  the  num 
ber  required. 
POSITION  is  di  ft  ingui  fried  into  two  kinds,  fingle 

and  double. 

» 

SEC     T.       L 

Of     SINGLE     POSITION. 

SINGLE  Pofition  is  when  one  quantity  is  re 
quired,  the  properties  of  which  are  given  in  the  quef 
tion. 

RULE. 

SUPPOSE  a  number  for  the  quantity  required,  and 
multiply  or  divide  it,  &c.  according  as  the  quantity 
required  was  multiplied,  divided,  &c.  then  -,  as  the 
refult  of  the  fuppofition,  is  to  the  fuppofition,  fo  is 
the  refult  given,  in  the  qfieftion,  to  the  number  re 
quired. 

EXAMPLES. 


EXAMPLES. 

To  find  fuch  a  number,  that  being  divided  by  2, 
4,  and  8,  refpeftively,  the  fum  of  the  quotients  fhall 
be  7. 

OPERATION.  f 

Suppofe  the  number  to  be  24,   then,  V  +  V  +  V  — 


Whence,  21  :  24  :  :  7  :  24X7  +  21  —  8,  /£*  number 
required. 

For,  4-+t+T-4+2+i=7;  therefore,  &c. 

A  man  having  a  certain  fum  of  money,  {aid  one 
half,  one  third,  and  one  fourth  of  it  being  added  to 
gether,  made  13  dollars  :  What  fum  had  he  ? 

Suppofe  he  had  36  do  1.  then  ^  -f  3T6  +  \6  =  1  8  +  1  2 
+  9—39,  which  ought  to  be  13,  by  the  queftion, 

therefore,  39  :  36  :  :  13  :  12,  the  anfwer.  , 

Three  men  found  a  purfe  of  dollars,  difputed  how 
it  fhould  be  divided  between  them.  A  faid  he  would 
have  one  third  ;  B  faid  he  would  have  one  third  anJ 
one  quarter  -,  well  fays  C,  I  fhall  have  but  2  dollars 
left  for  my  part  :  How  many  dollars  were  there  in 
the  purfe,  and  how  many  did  each  one  take  ? 

Supfofe  the  purfe  contained  1  1  dollars  : 

Then,  V+T  +VZ  =4+4+3  =  ii  - 
And,   1  2—  i  ii^ri,  which  ought  to  be  2. 
Wherefore,  i  :  2  :  :  12  :  24,  the  number  of  dollars  in 
the  purfe  ;  whence,  ^zrS,  the  number  of  dollars  that 
A  took;  and  ^-\-^=^-\-6^\^,the  number  that  Btcek. 

Delivered  to  a  banker,  a  certain  fum  of  money,  to 
receive  interefl  for  the  fame,  at  the  annual  rate  of 
6  dollars  per  cent  ;  at  the  end  of  7  years,  received 

for 


for  intcreft  and  principal,  2495  dollars  27^  cents  : 
What  was  the  fum  lent  ? 

Anfwer.    1736  doL  1 1  ~  cts. 

SECT.     II. 

Of    DOUBLE    POSITION. 

DOUBLE  Pofition  is  when  there  are  feveral  un 
known  numbers  in  the  queftion,  analogous  to  each 
other ;  fo  that  when  one  or  more  are  found,  the  reft 
may  be  had,  either  by  addition,  fubtraction,  or  mul 
tiplication,  &c.  according  as  the  queftion  requires. 

RULE. 

1.  ASSUME  two  convenient  numbers,  and  work 
with  them  as  the  queftion  directs,  finding  their  re- 
fults. 

2.  FIND  the  difference  between  thefe  refults  and 
the  refult  given  in  the  queftion,  and  call  thofe  differ 
ences  errors,  which  place  under  their  refpective  fup- 

pofitions.  ^r        C  x,  y,fup$ofttions. 

'    I  *>  &>  errors. 

3.  MULTIPLY  the  firft  error  with  the  fecond  fup- 
pofition  j  and  the  fecond  error  with  the  firft  fuppofi- 
tion.  Thusy  a  X  J,  find  b  X  #• 

4.  IF  the  errors  are  alike,  that  is,  both  too  great, 
or  both  too  fmall,  or  more  properly,  the  numbers 
from  whence  they  were  deduced,    are  both  either 
greater  or  lefs  than  the  true  ones,  you  muft  divide  the 
difference  of  the  products,   by  the  difference  of  the 

errors,  that  is,  aXy — £X#-r-tf— ^  ;  but  if  the  errors 
are  unlike,  that  is,  one  too  great  and  the  other  too 
fmall,  divide  the  fum  of  the  products  by  the  fum  of 

the 


the  errors  :  Thus,   ^X^+^x^-H  and  the  quo 
tient  in  either  cafe,  will  be  the  number  fought. 

EXAMPLES. 

A,  B,  and  C,  difcourfing  of  their  money :  Says  B, 
I  have  6  dollars  more  than  A  :  Says  C,  I  have  7 
dollars  more  than  B  :  Well  fays  A,  the  fum  of  all  our 
money  is  100  dollars  :  How  much  had  each  one  ? 

Suppofe  A  had  20  dol.  then  B  muft  have  20+6—26 
dol.  and  C  26  +  7=33.  <&/.  but  20+26+33=279, 
which  Jhould  be  100  by  the  queftion. 

Therefore,  100 — 79=121,  the  fir fl  error,  t  oof  mall. 

Again,  fuppofe  A  had  24  dol.  then  B  muft  have  24+6 
=•30,  <WC  30+7—37,  but  24+30+37=91,  which 
Jhould  be  100.  Therefore,  100 — 91—9,  thefecond  error, 
t  oof  mall. 

Whence,  *^*i^$&f=frMti&  of  the  fccondfuppo* 
Jltion  andfirft  error  ; 

And,  2oX9~i%Q~produftofthefrftfuffofttion  and 
fecond  error  ; 

Wherefore,   504 — 1 80-7-21— 91^27  dol.  A's  money ; 

Then,  27  +  6=33  dol.—B's  money,  ^^33  +  7—40 
C'j  money. 

A  man  having  been  to  market  with  hogs,  pigs  and 
geefe  ;  received  for  them  all  190  dollars,  for  every 
hog  he  received  4  dollars,  for  every  pig  75  cents, 
|and  for  every  goofe  25  cents  ;  there  were  for  every 
ipig  two  hogs  and  three  geefe  :  What  was  the  num 
ber  of  each  fort  ? 

Suppofe  he  had  1 2  pigs,   then  he  muft  have  24  hogs , 

**»  30  geefe'yby  the  queftion;  and  12 pigs  at  7 5  cts.  each, 

is  9  doL  i\hogs  at  4  dol.  each,  is  96  ^<?/.  and  36  £*£/<?  ^ 

.5  cts,  each,  is  9  *&/.  but  9  +  96+9^=114*  which  Jhould 

E  e  ^<? 


b  1 1 90 :  Therefore,  \  90 — 1141^76,  thefirft  err  or, too  fmalL 
Again  yfuffoje  he  hadi  6  pigsjhen  he  mufl  have  3  2  hogs> 
andafi  geefe ;  and  \6pigs  at  75  fAr.  is  12  *&/.  32  £0£.r 
0£  4  */0/.  /j  128  doL  and  48  ^^  tf/  25  r/j.  /j  12  ^/.  but 
12+128 +  121^:152  which  Jbould  be  190.  Therefore, 
190—152:1138,  thejeconderrvr,  toofmalL 

Whence  we  have  i6x76— i2X38-r76— 38— ?6o 
^.38^:20,  /^  number  of  figs y  and  20X^1140,  //?<? 
number  of  bogs -\  ^,20X3=60,  the  number  of  geefe. 


CHAP.     XIV. 

CONCERNING    PERMUTATION 

AND 

COMBINATION. 

S  E  C  T.      I. 
O/     PERMUTATION. 

PERMUTATION  is  the  changing  or  varying  the 
order  of  things ;  and  is  when  any  number  of  quan 
tities  are  given;  to  find  how  many  ways  it  is  poflible 
to  range  them,  fo  that  no  two  parcels  fhall  have  the 
fame  quantities  (landing  in  the  fame  place,  with  re- 
fpe6l  to  each  other. 

PROBLEM      I. 

To  find  all  the  variations  or  changes  that  can  be  made 
cf  any  number  of  things,  all  different  one  from  ano 
ther. 

FIRST  it  is  evident,  that  any  one  thingis  capable  of 
one  pofitiononly,,  and  therefore  cannot  poflibly  have 
any  change  or  variation  -,  but  any  two  quantities ;  as 

a 


C 


a  and  b,  are  capable  of  change  or  variatipn  ;  as  a,  b, 
and  b  ay  that  is,  the  number  of  variations  is  IX2* 
Again,  if  there  be  3  quantities  ;  as  #,  b,  c,  their  vari 
ations  are  a  b  c,  a  c  by  b  a  cy  b  c  a>  c  a  by  c  b  a  ;  for 
taking  only  the  two  firft,  a  and  b>  the  number  of 
their  variations  is  IX2>  therefore  taking  in  c>  the 
number  of  changes  is  1x2X3=6;  and  fo  on  for 
any  number  of  quantities.  Hence  we  have  the  fol 
lowing 

RULE. 

MULTIPLY  together  the  natural  feries  of  numbers, 
*>  2,  3,  4,  &c.  continually,  till  your  multiplier  is 
equal  to  the  number  of  things  propofed,  and  the  laft 
product  will  be  the  number  of  variations  required. 

EXAMPLES. 

In  how  many  different  pofitions  may  a  company  of 
8  perfons  ftand  ? 

Anfwer.  1X2X3X4X5X^X7X8-40320  pofi 
tions. 

How  many  changes  may  be  rung  with  12  bells  ? 
Anfwer.      1X2x3X4X5X6X7x8X9X10X11 

X  12—479001600,  the  number  of  changes  required. 

PROBLEM      II. 

To  find  all  the  poffible  alternations  or  changes  that  can 
be  made  of  ar.y  given  number  of  different  quantities,  by 
faking  any  given  number  of  them  at  a  time. 

THE  manner  in  which  this  problem  is  folved,  is  di~ 
reftly  thereverfe  of  the  laft  ;  for  it  is  manifefl,  that 
let  the  number  of  quantities  be  ever  fo  many,  and  we 
take  one  of  them  at  a  time,the  number  of  alternations 
be  equal  to  the  number  of  quantities,  Therefore 

it 


C      «o      ) 


it  follows,  that  the  operation  muft  begin  at  the  num 
ber  of  things  propofed,  and  then  decreafe  by  unity, 
till  the  number  of  multiplications  are  one  lefs  than 
the  number  of  things  propofed.  Hence  we  get  the 
following 


RULE. 

MULTIPLY  continually  together,  the  terms  of  the 
feries,  beginning  at  the  number  of  things  propofed  ; 
and  decceafing  by  unity  or  i,  until  the  number  of 
multiplications,  are  one  lefs  than  the  number  of  things 
to  be  taken  at  a  time,  and  the  laft  product  will  be  the 
number  of  alternations  required. 

EXAMPLES. 

How  many  different  pofitions  may  a  company  of 
9  men  be  placed  in,  taking  3  at  a  time  ? 

Here  the  number  of  multiplications  muft  be  2,  and 
the  feries  9,  8,  7,  6,  &c.  Therefore,  9X8X7  —  504, 
the  number  of  pofitions  required. 

How  many  alternations  will  the  letters  a  b  b  admit 
of,  taking  2  at  a  time  ? 

Anfwer.  3X2—6,  the  number  of  alternations  requir 
ed,  and  the  letters  willftand  thus,  a  h,  h  a,  ab>  b  a, 
bb>bb. 

How  many  alternations  or  changes  can  be  made 
v/ith  the  letters  a  b  c  d,  taken  3  at  a  time  ? 

Anfwer.  4X3X2—24,  the  num  ber  of  alternations 
required  \  and  the  letters  willftand 

{abcyacb>baC)bcaycal>)  cba~  alter,  of  ale 
lacdyadCiCadiCda^dac^dca—do.ofacd 
'      bcd>  bdc,  cbd>  cdb>  deb,  dbc^^do.  of  bed 

^=.do.  of  dab. 
How 


C 


How  many  alternations  or  changes  can  be  made  with 
the  letters  of  the  word  Algebra,    taking  4  at  a  time  ? 
Anfwer.     7  X  6  x  5  X  4=840* 

PROBLEM    III. 

¥0  find  all  the  alternations  or  changes  that  can  be 
made  of  any  given  number  of  quantities,  which  confjfl 
of  feveral  ofonejort,  and  feveral  of  another. 

RULE. 

1.  FIND  the  produft  of  the  feries,    1X2X3X4, 
&c.  to  the  number  of  things  to  be  changed,  which 
call  your  dividend. 

2.  FIND  all  the  alternations  that  can  be  made  of 
each  of  thofe  things  which  are  of  the  fame  fort,  by 
problem  i,  and  multiply  them  continually  together 
for  your  divifor. 

3.  DIVIDE,  and  the  quotient  refulting  will  be  the 
anfwer. 

EXAMPLES. 

Find  all  the  variations  that  can  be  made  of  the  fol 
lowing  letters,  a  a  b  c  c  c. 

OPERATION. 


^  1X2X3x4X5x6—  TIG—  number  of  va 

riations  that  can  be  made  of  6  different  things,  and 
i  X  2  z:  2,  tbe  variations  of  the  a's  ;  al/o,  i  X  2  X3=6 
the  variations  of  tbe  c's. 

Whence,  720-1-6x2.^60,  the  number  of  variations 
required. 

Find  all  the  different  numbers  that  can  be  made 
cf  the  following  numeral  figures,  1  1  1  22777. 

OPERATION, 


(        222,        ) 

OPERATION. 

Firft,  i^fi£y=&~rvariatiQns  of  the  iV,  and  1X2 
xzi^vtpations  of  the  z's  >  alfo^  ix^Xj— 6— van- 
it  ons  of  the  7*J. 

Whence,      1X2X3X4X5x6X7X8-5-6X2X7 
=40320-7-72:1:560,  theanfwer. 

SECT.      II. 
O/   COMBINATION. 

COMBINATION  of  quantities,  is,  when  any  number 
of  things  are  given,  to  find  all  the  different  forms  in 
which  thofe  quantities  can  be  poflibly  ordered,  and 
from  thence,  all  the  different  combinations  in  thofe 
forms,  without  any  regard  to  the  order  in  which  the 
feveral  quantities  ftand  in  thofe  combinations.  That 
is,  by  combination  we  determine  how  many  ways  it 
is  pofiible  to  combine  any  number  of  things,  fo  that 
no  two  combinations  fhall  have  the  fame  things  in 
both.  Combinations  of  the  fame  form,  are  thofe  that 
have  a  like  number  of  quantities  which  repeat  in  the 
fame  manner  in  both  :  Thus,  a  a  c  dy  and yy  x  z,  are 
of  the  fame  form>  but  aaa  be,  and s  mn  ry>  are  of 
different  forms. 

PROBLEM    I. 

fofind  all  the  different  combinations  that  can  be  made 
of  any  number  of  quantities  all  different  one  from  an- 
ctber,  by  taking  any  number  of  them  at  a  time. 
•t     THE  rule  for  the  folution  of  this  problem,    is  eafi- 
ly  deduced  from  the  rule  to   Problem  n,  of  permu 
tation,     For  it  is  plain,  that  the  number  of  combina 
tions 


(      223      ) 

tions  multiplied  with  the  changes  in  the  number  of 
things  taken  at  a  time,  gives  the  number  of  alterna 
tions  in  the  whole.  Therefore  it  follows,  that  the 
number  of  alternations  in  the  whole,  divided  by  the 
changes  in  a  number  of  things  equal  to  thpfe  taken 
at  a  time,  gives  the  number  of  all  the  different  com 
binations.  Hence  we  have  the  following 

RULE. 

1.  FIND  all  the  alternations  or  changes  of  the  giv 
en  quantities,  taken  as  many  at  a  time,  as  are  equal 
to  the  number  of  things  to  be  combined  at  a  time  5 
and  call  the  refult  your  dividend. 

2.  FrND  all  the  changes  in  as  many  quantities,  as 
are  equal  to  thofejto  be  taken  at  a  time  -,  and  call  the 
refult  your  divifot. 

3.  DIVIDE,  and  the  refulting  quotient  will  be  the 
number  of  combinations  required. 

EXAMPLES. 

• 

Find  all  the  different  combinations  that  can  be 
made  with  the  following  numeral  figures,  i,  2,  3,  4, 
5,  6,  taken  2  at  a  time. 

Here  the  number  of  given  quantities  are  6  ;  and 
the  number  to  be  taken  at  a  time  are  2  ;  therefore, 
6X5=30^=dividend  ;  and  i  X  2—2— divifor. 

Whence  30--- 2=115,  the  number  of  combinations 
required  -t  and  the  figures  will  (land  as  follows : 

12,  13,  14,  15,  16 

23,  24,  25,  26 

34,  35>  3-6 
4$>4<$ 

5*- 

FIND 


(       224      ) 

FIND  all  the* different  combinations  that  can  be 
made,  with  the  following  letters,  a  b  c  d  b>  taken  3 
at  a  time. 

Here  the  number  of  quantities  are  5,  and  the 
number  to  be  taken  at  a  time  are  3  ;  therefore,  5x 
4X3~6o— dividend  ;  and  1X2x3— 6~divifor. 

Whence,  6o-r-6mo,  the  number  of  combinations 
required  :  and  the  letters  will  ftand  as  follows  : 

a  b  C)  a  b  d,  b  b  b,  a  c  d 

a  c  b,  a  d  b,  bed 

b  c  by  bah 

€  a  b 

How  many  different  combinations  may  be  made 
with  the  following  numeral  figures,  i,  2,  3,  4,  5,  6, 
7 1  8,  9,  taken  5  at  a  time  ? 

Anjwer.    1*26  combinations. 

PROBLEM      II. 

¥0  find  the  number  of  different  combinations  tbat 
may  be  made  from  any  number  of  Jets  y  by  taking  ons 
out  of  each  fet  and  combining  tbem  together ;  the  things 
in  every  fet  being  all  different  one  from  another. 

RULE. 

MULTIPLY  the  number  of  things  in  each  fet  con 
tinually  together,  and  the  product  refulting,  will  be 
the  number  of  combinations  required. 

EXAMPLES. 

How  many  different  combinations  of  two  letters, 
may  be  made  of  thefe  two  fets  an  w  and  s  x y  ? 

Here 


Here  the  number  of  things  in  each  fet  are  3  r 

Therefore,  3X3^9,  the  number  of  combinations 
required. 

The  method  of  making  the  combinations,  may  be 
{hewn  in  the  following  manner. 

Write  down  the  two  fets  one  beneath  the  other, 
and  join  thofe  letters  that  are  to  be  combined,  with 
iftraight  line, 

~  a  n  iv 

i  i  i 

s  x  y 

Then  drawing  lines  from  s  to  a,  from  x  to  »,  and 
from  y  to  wy  you  will  have  three  of  the  required 
combinations,  to  wit,  s  a,  x  ny  and  y  w< 

Again,  let  the  fets  be  placed  as  before  : 


I 


y 

Then  joining  s  and  wy  x  and  a,  andj  to  ny  we 
get  s  wy  x  a  and^  n.  Once  more,  place  the  fets  as 
ibove . 


s         x'       y 
Then  joining  s  and  ny  x  and  tu,  and  y  to  tf,  v;t 
get  s  ny  x  wy  and  y  a. 
Hence,  all  the  combinations  are  as  follows., 

x  ay  x  n,  x  w, 

F  f  Suppofc 


(        226        ) 

Suppofe  there  are  three  flocks  of  flieep  ;  in  one  of 
which  there  is  10,  and  in  the  other  two,  20  each  : 
To  find  how  many  ways  it  is  poflible  to  choofe  3 
fheep,  one  out  of  each  flock. 

Thus,  10X20X20=4000,  the  anfwer. 

PROBLEM     III. 

7 0  find  the  number  of  forms  in  which  any  given 
number  of  quantities  may  be  combined,  by  taking  any 
number  at  a  time  ;  wherein  there  arejeveral  of  one  fort, 
and  fever  al  of  another. 

RULE. 

1.  WRITE  the  quantities  according  to  the  order 
of  the  letters.          Ttttfc  ay  a,  by  cy  d. 

2.  JOIN  the  firft  letter  to  the  fecond>  third,  fourth, 
&c.  to  the  lad  -,  and  the  fecond  letter  to  the  third, 
fourth,  &c.  to  the  laft  ;  alfo,  the  third  letter  to  the 
fourth,  fifth,  &c.  to  the  laft  :    Proceeding  in  like 
manner  through  the  whole,  taking  care  to  reject  all 
combinations  that  have  before  accrued  a  and  you 
will  have  the  combinations  of  all  the  twos. 

3.  JOIN  the  firft  letter  to  every  one  of  the  twos, 
and  the  fecond,  third,  fourth,  &c.  in  like  manner 
to  the  laft ;   and  you  will  have  the  combinations  of 
all  the  threes. 

Thus,  a  a  a,  a  a  by  a  a  cy  aa  d,  ab  ey  a  b  dy  a  c  dy 
b  a  a,  a  b  £,  b  a  cy  b  a  d>  bb  c>  b  b  dy    bed, 

c  a  ay c  c  ay c  c  by  — c  c  dy 

da  ay  —  — —  dd  ay _  d  d  cy 

And  proceed  in  this  manner,  till  the  number  of 
things  in  the  combination,  are  equal  to  the  number 
to  be  taken  at  a  time. 

Note.  All  thoje  combinations  'which  contain  more 
things  of  the  fame  fort ,  than  are  given  of  the  like 
kind -in  the  queftion,  muft  be  rejected.  EXAM* 


(227        ) 


EXAMPLES. 

Find  all  the  different  forms  of  combination,  that 
can  be  made  of  the  letters  a  a  b  b  c  c}  taken  4  at  a 
time, 

OPERATION. 

a  ay  a  by  a  c>  b  by  b  cyc  c~  combinations  of  the  twos. 
a  a  by  a  a  Cy    b  b  ay  b  a  c}    bb  cy  b  c  Cy   ac  Cy~  combi 
nations  of  the  threes. 

a  abby  a  ab  Cy  b  b  c  ay  c  c  a  by  aa  c  (>  b  b  c  Cy  iz 
combinations  of  the  fours. 

Whence  y  aabby  b  b  c  cy  a  a  c  cy  and  a  a  c  by  b  b  a  c , 
c  c  a  by  are  the  two  forms  required. 

%ind  all  the  different  forms  of  combination  that 
can  be  made  of  the  following  figures,  22334455,  ta 
ken  3  at  a  time. 

OPERATION, 

Thusy  22,  13,  24,  25,  33,34,  35>  44>45>5S- 
combinations  of  the  twos. 
223,  224,  225,  234,  235,  245,  233,  334,  335, 

345>  244,  344*  44S>  255>  355>  455  —  combinations 
of  the  threes. 

Whencey  223,  224,  225,  233,  433,  533,  244, 
344,  544,  255,  355,  455>  ^234,  235,  245,  345, 
are  the  forms  required. 

THUS  far,  concerning  Permutation  and  Combin 
ation. 


(        223        ) 


CHAP.      XV. 

Of    INVOLUTION. 

WHEN  any  number  is  multiplied  into  itfelf, 
and  that  product  multiplied  with  the  fame 
number;  and  fo  on,  it  is  what  is  called  Invoiution, 
and  the  feveral  produ&s  refujting,   are  called  the 
powers  of  the  multiplying  quantity,   or  root.  Thus* 

3X37  3X3X3>  3X3X3X3,  &c.  are  the  powers  of 


3.    Aad  generally,  a^a,  ^X^X^,   and 

&c.  are  the  powers  of  a  ;  whofe  height  is  denomina 
ted  by  the  number  of  multiplications  more  one. 
HENCE,  the  2d  power  of  ip,  is  loXiomoo 
the  3d 


the  4th  -  ioXioXioXio=iQQOO. 
Therefore  it  follows,  that  the  powers  of  any  quan 
tity,  are  aMeries  of  numbers  in  Geometrical  Propor 
tion  continued,  whofe*  firft  term  and  ratio  is  the  fame, 
to  wit,  the  root  of  the  power  :  Confequently  the 
height  of  the  power  at  any  particular  term,  will  be 
pcprelfed  by  the  exponent  of  that  term  :  AS  in 

12  3  4  &c>  Expon. 

?**     10,   10X10,  1,0x10X10,   lo 


HERE  it  is  evident,  that  the  index,  or  exponent 
of  each  term  of  the  Geometrical  feries,  is  equal  tc 
the  number  of  multiplications  of  the  firit  term  witl 
itfelf,  to  that  place,  more  one,  and  is  therefor^  call 
ed  the  index,  or  exponent  of  the 


Thus  ---- 

1  5x5x5X5X5^:3125-5^^^^/5: 

on  for  others. 

WHENCE 


WHENCE  it  follows,  that  to  raife  any  number  to 
any  given  power,  is  no  more  thart  to  multiply  the 
given  number  into  itfclf,  fo  often  as  there  arc  units 
in  the  index  of  the  power  —  i. 

EXAMPLES. 

Required  the  5th  power  of  9. 
OPERATION. 

9 
9 

%i=.id  power  of  9 
9 

—  3d  power  cf  9 
9 


656  1  =4/£  fewer  of  9 
9 


—  $tb  power  of  y,  as  requlr. 

Required  the  7th  power  of  8. 
Thus,    8X8X8X8x8X8X8  =  2097152  =  7/4 
power  0/8. 


E 


CHAP.     XVI. 

Of    EVOLUTION. 

VOLUTION  is   the  converfe  of  Involu 
tion  j  and  is  when  any  power  is  given,  to  find 

the 


the  number  from  whence  fuch  power  was  produced, 
which  number  (as  we  before  faid)  is  called  the  root 
of  the  power  -,  and  the  bufinefs  of  finding  it,  is  called 
extraction  of  roots. 

ALL  powers  whatever,  are  produced  by  the  contin 
ual  multiplication  of  their  roots  into  themfelves,  as 
is  evident  from  what  has  been  faid  ;  yet  there  are 
many  powers  which  have  no  finite  root,  that  is,  whofe 
true  and  adequate  root  cannot  be  expreffed  in  finite 
terms ;  but  by  approximation  may  be  determined  to 
any  affigned  degree  of  exaclnefs. 

THESE  powers  are  called  furds,  or  irrational 
powers.  > 

PROBLEM    I. 

2*o  extra  ft  the  root  of  tbejquare  orjecond  -power  of 
any  number. 

RULE. 

1.  PREPARE  the  given  nurpber  for  extraction,  i.  e, 
diftinguifh  it  into  periods  of  two  figures  each,  by  be 
ginning  at  the  unit's  place  and  placing  a  point  over 
the  firft,  third,  fifth,  &c.  figures  of  the  given  num 
ber,  and  if  there  are  decimals,  point  them  in  the  fame 
manner,  from  unity  towards  the  right  hand. 

2.  FIND  a  number  by  the  help  of  a  table  of  pow 
ers,  whofe  fquare  is  equal  to,  or  lefs  than  the  firft  pe 
riod  on  the  left  hand,  and  this   number  will  be  the 
firft  figure  of  the  root,  which  place  in  the  form  of  a 
quotient  -3  then  fubftracV  its  fquare  from  the  afore  - 
faid   period ;  and   to  the  remainder  annex  the  next 
period  for  a  dividend. 

3.  DOUBLE  the  firft  figure  of  the  root  for  a  divifor. 

4.  FIND  fuch  a  quotient  figure,  that  when  annex 

ed 


ed  to  the  divifor  and  the  refult  multiplied  with  the 
fame  number,  the  produft  will  be  equal  to,  or  lefs 
than  the  dividend  j  and  this  will  be  the  fecond  figure 
of  the  root. 

5.  To  the  remainder  annex  the  third  period  for  a 
new  dividend^  and   add  the  figure  in  the  root  lad: 
found  to  your  former  divifor  for  a  new  one. 

6.  FIND  the  third  figure  of  the  root  as  you  found 
the  fecond  ;  and  fo  on,  till  all  be  done. 

Note  i ,  -  If  there  is  a  remainder  after  all  the  periods 
are  annexed,  the  given  number  is  a  furd,  and  you. 
muft  approximate  to  the  root,  by  annexing  cyphers 
two  at  a  time,  to  the  remainder, 

2.  If  the  given  number  confifts  of  integers  and 
decimals,  you  muft  ponit  off  as  many  places  in  the 
root,  as  there  were  periods  of  decimals,  in  the  given 
number. 

EXAMPLES. 

Required  the  fquare  root  of  58081. 

OPERATION. 

'      •    •    . 

58081(241 

4 
\fl  divifor— 4.4)  i%o 

4   i?^ 

id  divifor-=4&  i )    481 
481 


therefore,  241  is  the  root  required,  as  may  be  proved 
ly  involution  :  Thus,  241X241  —  58081,  which  is  the 
fame  as  the  given  number  :  Whence >  &c. 

Required 


Required  the  fquare  root  of  1000. 
OPERATION. 


1000(31.622  fcfc.:=:  root  required, 
9 

61)100 
i     61 

626)39.00 
6  37S6 

6322)14400 

V2    12644 

63242)175600 

2    126484 

fifc.  - 

49116  &V. 

Required  the  fquare  root  of  105462.5625  : 


OPERATION. 


(      233      ) 

OPERATION', 

105462.5625(324.75  —  rev! 
9 

62)154 
a  124 

644)3062 
4  2576 

6487)48656 
7  45409 

64945)324725 
324725 

o 
P  R  O  B  L  E  M  II. 

!T0  extract  thefquare  root  of  a  Vulgar  Fraction* 
RULE. 

EXTRACT  the  root  of  the  numerator,  for  the  nume* 
rator  of  the  root ;  and  the  root  of  the  denominator, 
for  the  denominator  of  the  root. 

EXAMPLE. 
Required  the  fquare  root  of  T£|4- 


OPERATION, 


(       234      ) 


OPERATION. 

225(15:1:  numerator  of  the  root. 
i 

1024(32— denominator  of  the  root, 
"5  1_ 

62)      124 

124 

,  y~  is  the  root  required, 
PROBLEM    III. 

¥0  find  the  root  of  the  third  power  or  cube,  by  af~ 
proximation. 

RULE. 

1.  DISTINGUISH  the  given  number  into  periods 
of  three  figures  each,  by  beginning  at  the  unit's 
place,  and  placing  a  point  over  the  firft,  fourth,  fe- 
venth,  figures,  &c.  and  if  there  are  decimals,  point 
them  from  the  unit's  place  towards  the  right  hand, 
in  the  fame  manner. 

2.  FIND  the  root  of  the  firft  period  on  the  left 
hand,  by  the  help  of  the  table  of  powers,  and  annex 
to  it,  as  many  cyphers  as  there  are  remaining  periods, 
then  involve  this  number  to  the  fame  power  as  the 
given  number,  and  call  the  refult  the  fuppofed  cube ; 
then  :  As  twice  the  fuppofed  cube  +  the  given  cube  j 
is  to  twice  the  given  cube  -f  the  fuppofed  cube  $  fo 
is  the  root  of -the  fuppofed  cube ;  to  the  root  requir 
ed,  nearly. 

.3.  IF  a  greater  degree  of  exa&nefs  is  required,  in 
volve  the  root  already  found,  to  the  third  power,  and 

call 


call  the  refult  the  fuppofed  cube,  with  which  pro 
ceed  as  as  before,  and  fo  on,  to  any  degree  of  exa<5b- 
nefs. 

Note.  When  the  root  is  finite,  you  mayfometimes 
Jave  the  trouble  of  repeating  an  operation,  by  in- 
creafing  the  right  hand  figure  of  the  root  found,  by 
unity. 

EXAMPLES. 

Find  the  cube  root  of  1367631. 
OPERATION. 

Firft,  1  3  67  63  1  is  the  given  number  prepared  for  ex- 
traftion,  the  root  of  whofe  firft  period  (i)  is  i  ;  then 
iooX  iooX  tQQ=iOGOGGQ~&04  cube  ;  and, 


^1000000X2+1367631  :  1367631X2+1000000:: 
100,     /'.  e.    3367631  :  3735262  ::  100 

100 


3676310  iu~  root  requtr, 
3367631 


3086790 
Required  the  cube  root  of  729001101. 

Firft,  729001101  is  the  given  number  pointed,  and 
the  root  of  the  fir  ft  period  (7  29) —9  ;  therefore  900  x 
900X 900—7 -lyoQcvoo—Juppofedtubei  then, 

as    729000000X2+729001101    ;  729001101X2  + 
72900000  o :;  900, 

That 


<?bat  is,  2^87001101  :  2187002202::  900 

900 


2187001101)1968301981800(900,0004— 
19683009909       \rbotnearLy. 

9909000000. 

THE  cube  root  of  a  Vulgar  Fraction,  is  found  by 
cxtra&ing  the  root  of  the  numerator  and  denomina-r 
tor. 

PROBLEM    IV, 

'/a  sxtraft  the  roots  of  powers  in  general. 
RULE. 

1.  LET  the  index  of  the  power  whofe  root  is  tp 
be  extracted,  be  denoted  by  •#. 

2.  POINT  the  given  'number  into  periods  of  as 
many  figures  each,  as  there  are  units  in  #,  begin 
ning  at  the  unit's  place  3  and  if  there  are  integers  and 
decimals  together,  let  them  be  pointed  both  ways 

from  unity. 

3.  FIND  the  root  of  the  firft  period,    by  the  help 
of  the  table  of  powers,  and  this  will  be  the  firft  fi 
gure  of  the  root. 

4.  SUBTRACT  the  n  power  of  the  firft  figure  of  the 
root,  from  the  firft  period,  and  to  the  remainder  an 
nex  the  firft  figure  of  the  next  period^  which  refuH 
call  your  dividend. 

5  INVOLVE  the  root  now  found  to  the  » — i  power, 
and  multiply  the  refill  t  with  n  for  your  divifor. 

6.  DIVIDE,  and  the  quotient  will  be  the  fecond 
figure  of  the  root. 

7.  INVOLVE  all  the  root  now  found  to  the  n  power, 
Ind  fubtract  it  (  always  )  from  as  many  periods,  as 

you 


you  have  found  figures  of  the  root :  But  if  the  num 
ber  to  be  fubtraded,  is  greater  than  the  aforefaid  pe 
riods,  the  laft  figure  of  the  root  is  too  great,  which 
mult  therefore  be  diminilhed,  fo  that  the  n  power  of 
the  root  now  found,  may  be  taken  from  the  aforefaid 
periods. 

8.  To  the  remainder  annex  the  firft  figure  of  the 
next  period  for  a  new  dividend,  then  find  a  new  di- 
vifgr  as  before  -,  and  fo  on,  till  the  whok  be  done. 

EXAMPLES. 

Required  the  cube  root  of  61209.566621  : 
OPERATION. 

Here  n  — 3>  therefore  the  given  number  fainted  is 

61209.566621,  and  the  neareft  root  of  the  fir  ft  -period 
(6 1 )  is  3,  which  is  thefirft  figure  of  thereof,  the  n  pow 
er  of  which  is  3X3X3=27  -,  and  61  —  27  —  34,  which 
having  the  fir  ft  figure  of  the  next  period  annexed  to  if, 

becomes  342:1:  fiv 'ft  dividend \  tf#*/ 3X3X3  — 27  =r/foy/ 
divifor :  Whence,  27  )342(9~  fecond  figure  of  the  root> 
and  the  whole  of  the  root  now  found  is  39  j  therefore, 
39X39X39=59319=  n  power  of  39,  which  being 
Jub  traced  from  the  two  firft  periods,  leaves  1890,  and 

\*y*$=.Jecond  dividend;  aljo,  39X39X3— A-S^S  — 
Jecond  divifor  •>  whence,  4563)18905(4—  third  figure 

oftheroot.  dgain,  394X394X394z::^i  \6-2$%4,whicb 
Jubtrafted  from  the  three  firft  periods,  leaves  46582, 

then,  465826=:  third  dividend,  'and  394X394X3— 
465708^=^  third  divifor  -y  whence,  465708)465826(1 
—fourth  and  I  aft  figure  of  the  root,  and  becaufe  there 
are  two*periods  of  decimals  in  the  given  number,  the 
root  required  is  39.41  ;  for  39.41X39.41X39.41  = 
61209.566621^  the  number  whofe  root  was  required: 
&c.  Required 


C 


Required  the  6th  root  of  148035889. 
OPERATION. 

Firft,  txtraft  thefquare  root,  and  then  the  cube  roof 
cf  that  rejnlt  will  give  the  root  required  : 

Thus,  148035889(1216? 
i 

22)    48 

2     44 

241)  403 

I      241 

2426)  16258 
6  I4556 

-4327)170289 
170289 


w  » 

Again,  1  2  1  67  (  23=  root  required, 

2X^X2—8 


23X23X23=12167 

o 
The  fame  at  one  operation  : 

Thus,  148035889(23 
2X2X2X2X2X2-64 

2X2X2X2X2X3=96)840 
23X23X23X23X23X23  =  148035889 

•  '  IN 


(      239      ) 

IN  extradling  the  roots  of  heigher  powers,  it  will 
be  beft  to  extraft  fquare  root  out  of  fquare  root  fuc- 
ceilively,  as  often  as  the  index  of  the  given  power  is 
divifible  by  2  :  Thus,  in  the  i6th  power,  the  index 
(i6)is  divifible  by  2,  four  times  -3  for  i6~-2n:8,  8-f- 
2—4,  4-~-2~2>  and  2-7-2—1  :  Whence  it  follows, 
that  the  root  of  the  i6th  power  may  be  obtained  by 
four  feveral  extraflions  of  the  fquare  root  5  and  the 
like  may  be  fhcwn  of  all  the  even  powers. 


THE  END  OF  BOOK  FIRST. 


BOOK      II. 

OF     ALGEBRA, 

*&*Oto&^i*^^ 

CHAP,       1. 

6f     DEFINITIONS 

AND 


ALGEBRA,    one  of  the   mod  important 
branches  of  mathematical  fcierice,is  a  method  of 
computation  by  figns  and  fymboh,  which  have  been 
invented  and    found  ufeful  for  that  purpofe.     Its 
,  invention  is  of  the  higheft  antiquity,  and  has  juft- 
|  ly  challanged  the  praife  and  admiration  of  the  learn 
ed  in  all  ages;     Arithmetic  is  indeed  ufeful,  and  is 
not  to  be  the  lefs  valued,  becaufe  it  is  allowed  to  be 
the  moil  clear  and  evident  of  the  fciences  j  yet  it  is 
confined  in  its  object,-  and  partial  in  its  application, 
Geometry  for  clearnefs  of  principles^  and  elegance 
of  demftnftration,  no  lefs  deferves,  than  commands 
our  efteem;    but  the  many  beautiful  theories,  that 
srife  from  the  application  of  Algebra  and  Geometry 
to  each  other,  fully  evince  the  excclUncf  aind  etften* 

Hh  fiveitefs 


fivenefs  of  the  former.  The  doctrine  of  Fluxions, 
Which  is  eftcemed  the  fublimity  of  human  fcience^  de 
pends  on  the  noble  fcience  of  Algebra  for  its  exift- 
ance  and  application.  In  a  word,  Algebra  is  juftly 
efteemed  the  key  to  all  our  mathematical  inquiries. 

IN  Algebra,  like  quantities  are  thofe  which  have 
the  fame  letters  :  Thus,  ax  and  ax  are  like  quantities; 
but  ax  and  dx  are  unlike  quantities. 

GIVEN  or  abiblute  numbers,  are  thofe  whofe  val 
ues  are  known  :  Thus,  6,  7,  &c.  are  given  numbers, 
becaufe  their  refpeftive  values  are  known  ;  but  the 
quantities  x,  y3  &c,  are  not  given  quantities,  becaufe 
their  values  are  not  known,  and  are  therefore  called 
unknown  quantities. 

SIMPLE  quantities  are  fuch  as  have  but  one  term  : 
Thus,  £,  axby  and  xyz,  are  fimple  whole  quantities, 

and~r-  and—  7-  are  fimple  fractional  quantities. 

COMPOUND  quantities  are  fuch  as  confift  of  feveral 
terms  connected  by  the  figns-f  and  —  :  Thus,  a-\- 
Jt-^c  —  </and  ax  —  xy  are  compound  whole  quantities, 

.  a     c     dx     a-\-b  .  c    -.       . 

and  7+-.  —  r-  i.  —  —7  are  compound  fractional  quan- 
b     d     b      c  —  d 

tities.     Compound  quantities  ^have  fometimes  aline 


drawn  over  them  ;  as  a^b-^c  —  d. 

CO-EFFICIENTS  are  numbers  prefixed  to  quantities, 
denoting  how  many  times  the  quantity  to  which  they 
are  prefixed,  ought  to  be  taken  :  Thus,  3  a  denotes 
that  the  quantity  a  is  to  be  taken  3  times  $  alfo,  na 
ihews  that  the  quantity  a  is  to  be  taken  as  many  times 
as  there  are  units  in  n  :  Therfore,  co-efficients  mul 
tiply  the  quantities  to  which  they  are  prefixed  ;  and 
quantities  which  have  no  co-efficient  prefixed  to  them, 
are  always  underftood  to  have  an  unit  for  their  co 
efficient  :  Thus,  a  is  i  a,  x  \  #,  &c. 

A 


A  POSITIVE,  or  an  affirmative  quantity,  is  a  quan 
tity  having  the  fign  -j-  before  it  ;  as  -\-a  :  Alfo,  all 
quantities  that  have  no  figns  fet  before  them,  as  the 
leading  quantity  generally  hath  none,  are  underftood 
to  have  the  fign  4-,  and  are  therefore  called  pofitive 
quantities. 

WHEN  quantities  have  the  fign  —  before  them, 
they  are  called  negative  quantities  :  As  —  a,  —  x  ; 
and  when  any  quantity  is  to  be  diftinguilhed,  as  a 
quantity  to  be  fubtra&ed,  the  fign  —  muft  be  placed 
immediately  before  it. 

QUANTITIES  are  faidto  have  like  figns,  when  t^iey 
are  all  +  or  all  —-, 

UNLIKE  figns  is  when  the  figns  are  +  and  —  . 

A  QUANTITY  confiding  of  two  terms,  as, 


is    called  a  binominal  $  a+b+c>    a    trinominal 


y  a  quadrinominal,  &c. 
A  RESIDUAL  quantity,  is  the  difference  of  two 

quntities,    Thus,  a  —  £,  is  a  refidual  quantity. 

THE  letters  made  ufe  of  to  reprefent  the  unknown 
quantities,  are  thofe  of  the  laft  part  of  the  alphabet^ 
and  the  letters  of  the  firft  part,  reprefent  thofe  that 
are  known. 

THE  principal  figns  by  which  quantities  are  man 
aged  in  Algebra,  are  the  following,  in  addition  to 
thofe  made  ufe  of  in  the  firft  book  of  this  treatife, 

and  Explanations. 

is  the  fign  of  the  fquare  root. 
-  -  of  the  cube  root. 
.  of  the  n  root. 

.  of  more  or  lefs. 


I 

x  or  *T  denotes  the  fquare  ropt  of  x* 
x  or  #T  the  cube  root  of  AT. 


or  <*+£)  T  the  fquare  root  Qfa+£ 
or  ^4"  the  «  root  of 


*•  the  reciprocal  of  #. 


the  reciprocal  of-. 
»  y 

<i±^  the  (urn  or  difference  of  ^  and  £t 

AXIOM  S. 

1.  IF  to  thofe  quantities  that  are  equal,  there  be 
gdcjed  the  fame  quantity,  their  fum  will  be  equal. 

2.  IF  from  thofe  quantities  that  arc  equal,  thertf 
be  taken  the  fame  quantity,  the  remainders  will  be 
equal. 

3.  IF  thofe  quantities  which  are  equal,  be  multi 
plied  with  the  ftme  quantity,  their  products  will  be 
equal. 

4.  .IF  thofe  quantities  that  are  equal,  be  divided 
by  the  &me  quantity,  .the  quotients  will  alfo  be 
equal. 

5.  Two  quantities  refpeftiyely  ecjual  to  a  third,, 
are  equal  to  each  other. 

6.  EQUAL  powers,  or  roots  of  equal  quantities^ 
are  equal  to  each  other. 

7.  IF  to  any  whole  number,  there  be  added  any 
other  whole  number,  the  fum  will  be  a  whole  num- 
ker.' 

8.  IF  from  any  whole  number,  there  be  taken  any 
other  whole  number,  what  remains  will  alfo  be,  a 
whole  number. 

9- 


(      345      ) 

g,  IF  any  whole  number  be  multiplied  with  any 
p$her  whole  number,  the  pro4uft  will  alfo  be  a  whole 
number. 


CHAP.     II. 

ADDITION  of  WHQLE 
A    D  D  I  T  I  O  N  cpnfifts  of  three  cafes. 

CASE    i. 

Wbcn  the  quantities  arc  alike,  and  have  'li 
RULE. 

ADD  the  co-efficients  together,  and  to  their  furn 
annex  the  common  quantity,  prefixing  the  common 
Jign, 

EXAMPLES. 


lab      —6^     —  -  2xy       zx  —  2  a 
tab       —    x     —  iQx        6x  —    ^ 


—  ixy       ax  —    a 
-^    xy          x  —   a 


IT  ak      —  >8#    —  i%xy 


_ 

av  —  3  xy'i  +  iaz—    b  —  4^;*  —  6  0+3—2^ 

t^ 

tf<tf—    #jy*-f3  <*z  —  3  ^  —  ^  w*  —  2  *+  *—  *8  d 

t^ 

av  —  6  #>*-    #£-—  •    £  —  WT-—    #-0—   d 


1  3  av—  -i  o^y  *  +  1  4^%—  5<£  —  ^i  i  w  *  •—  9#  +  4—1 

CASE      II. 

^Z>^»  /Ad-  quantities  are  alike,  but  have  unli 
RULE, 

1.  ADD  all  the  affirmative  quantities  into  one  fum 
by  the  Jaft  rule,  and  the  negative  into  another. 

2.  SUBTRACT  their  co-efficients,  the  lefs  from  the 
greater,  and  to  their  difference,  prefix  the  fign  of  the 
greater,  annexing  the  common  quantity. 

THE  reafon  of  the  foregoing  rule  will  appear  evi 
dent,  if  you  put  a  =  debt  due  to  B,  and  —  a  the 
want  of  a  debt,  or  a  debt  due  from  B  j  then  the  bal 
ance  is  evidently  equal  o,  or  4-  &  —  0  ^  o  :  Whence,. 

EXAMPLES, 


3  *  +  6  J> 
S*+2j 
io  ^ — 


—  io  ay  ~ yo»  <?/<?  negatve.         — 10  — 
•{-    5  *?  ^-Jum  of  the  affirmative.     +  7  ^  4- 


^—   5  #y  ~Jum  required, 


— 4  </* — a  //  **4-jE  —3  *  4-4  ^  —  w* 


—8  </*-- 

CASE      III. 

^T^/r  the  quantities  are  unlike,  and  have  unlike  Jlgns. 
RULE. 

WRITE  the  quantities  one  after  another  with  their 
proper  figns,  and  they  will  be  the  fum  required. 

Note.  If  there  be  like  quantities  given,  you  muft 
colleft  them  by  the  preceding  rules. 

EXAMPLES. 


7  V  +99? 
—4^4.7  c 


—  8  d  <  7  <y 

—3  a  4-4  1  —2  c  —  2  d  ^=.Jn 


4 

I  X 

^-^^jy"57-—  2  /fr  +2J*5"      —  c&%  4-96  —  3  \/^*~ 


*  —  rf*~-s  -/a* 


CHAR 


CHAP,     III. 

SUBTRACTION  of  WHOLE 
TITIES. 


A 


LGEBRAIC  Subtra&ion is  performed  by 
the  following  general 

R  U  L  E. 


CHANGE  the  flgns  of  the  quantities  in  the  fubtri- 
hend  (or  fuppofe  them  in  your  mind  to  be  changed) 
then  add  the  quantities  with  their  figns  changed,  to 
the  number  from  which  fubtra&ion  is  to  be  made, 
by  the  rules  of  the  Jaft  chapter  and  their  fum  will 
be  the  remainder  required. 

THE  reafon  of  this  rule  will  appear  obvious,  when 
we  confider  that  fubtraftion  is  the  revrfe  of  addition  j 
and  therefore}  to  fubtraffc  an  affirmative  or  negative 
quantity,  is  the  fame  thing  as  to  add  its  oppofite 
kind  :  Whence,  if  —  a  is  to  be  taken  from  +  tf,  the 
difference  will  be  +  2  a>  for  if  the  remainder  2  a  be 
added  to  the  fubtrahend  —  *z,  their  fum  will  be  3:  a 
n  the  number  from  which  fubtra&ion  was  made  i 
Whence,  &c« 

EXAMPLES. 


From  44       4lu^~%b* 

3  a       ibu  —  2^a          4«y+ 


Remains  a       y,bu~~>  b* 

34  V 


(      249      ) 


Ic— 

Ir  any  doubt  arife,  refpeding  the  truth  of  the 
peration,  add  the  remainder  to  the  fubtrahend, 
rhich  fum  muft  be  equal  to  the  other  number. 


CHAR      IV. 

Of     MULTIPLICATION. 

LGEBRAIC     Multiplication    confifts  of 
three  cafes. 

CASE    I. 

When  loth  the  faftors  are fm fie  quantities. 
RULE, 

MULTIPLY  the  co-efficients  together,  and  to  their 
roduct  annex  all  the  letters  in  both  faffcors,  as  in  a 
rord ;  this  exprefTion  being  wrote  with  its  proper 
gn,  will  give  the  product  required. 

Note.  Like  figns  give  +,  and  unlike  Jigns — for 
the  produft, 

I  i  EXAMPLES* 


(      25°     ) 


EXAMPLES. 


4.  30  3«         —  21  j^   —  3 

—  60  2       —  2 


—  1  8  tftffo     —  4-1  yyy  -f-6  tfzejyjp  produff. 

CASE     II.    ^ 

#£  of  the  factors  is  a  compound  quantity. 

R  U  L  E. 

1.  WRITE  the  compound  quantity  for  the  multi 
plicand,  and  the  fimple  quantity  for  the  multiplier. 

2.  OBTAIN  the  product  of  the  multiplier  with 
every  particular  term  of  the  multiplicand,  by  the  lad 
rule,  and  place  the  terms  of  the  product  one  after 
another,  with  their  proper  figns,  found  as  in  the  laft 
rule,  and  you  will  have  the  product  required. 

EXAMPLES. 

a  4-  b        3  ab  -J-  cd  2  aa  +  2  ab  -{-  It  ' 

a  d  ia 


au  —4  cv  +34  2.7  ddd  —  aaa 


— 3  auy^ii  cvy  —io2y     8 1  dddw  —3  aaaw. 
CASE    III. 

When  both  tbefySiors  arc  compound  quantities. 

RULE. 


R  ULE. 

MULTIPLY  every  particular  term  of  the  multiplier, 
with  all  the  feveral  terms  of  the  multiplicand,  as  in 
the  laft  rule,  the  feveral  products  collected  into  one 
fum  by  the  rules  of  addition,  will  give  the  who!? 
product  required. 


+ 

+ 


y 

y 


EXAMPLES, 
a—   b 


V  —    2Z 
V+    22 


vv  -f-  vy 


+yy 


aa-  —  ab 

ab 


OT  —  2  -VZ 

bb        -f  2  vz  — 


vv  -\-ivy  +yy 

yy  -f  xx 

yy  —  xx 


yyyy 


aa 


—  bb 


2X —  I 


—  yyxx  —  xxxx 


2  xx  —  8  x 

•*-  2  xy  — 


yyyy 


xxxx 


4  xxy 


<  —  2  #j—  9  A:  4-  4 


HAT  -f  X  —  or  -r-  x  +  gives  — ,  an'd^ —  X 
—  gives  -j-  for  the  product,  is  demonftrable  feveral 
ways,  but  none  more  fimple  than  the  following.  Sup- 
pofe  a  ~  b  j  then  a — b  zzo  :  Now  it  is  plain,  that  if 
this  exprefllon  be  multiplied  with  any  number  what 
ever,  the  product  will  be  ~o  :  Therefore,  fuppofe 
2 — b  =o,  is  to  be  multiplied  with  -f- n  >  now  'lt  is 
manifeft,  the  firfl  term  of  the  product  a  X  n  will  be 
poiitive  j  or  -f-  ##>  becaufe  bo;h  the  fadors  are  pofi- 


(     252     ) 

tive  y  confequently  the  other  term  of  the  product 
X  —  b  muft  be  negative,  or  —  nb ;  for  both  terms 
of  the  prod  uft  taken  together,muft  deftroy  each  other, 
and  their  amount  =o  ^Jthat  is,  na  —  nb  zi  o  :  Gon- 
fequently  ~j-  X  •—>  or  —  X  +  gives  —  for  the  prod- 

vft. 

AGAIN,  fuppofe  a  —  b  —  o,  be  multiplied  with 
—  n ;  the  firft  term  of  the  product  —  n  x  a  will,  be 
negative,  or  — na>  by  what  has  been  proved  :  Con 
fequently,  the  other  term  —  #  X  —  b  will  be  pofitive, 
or  -j-  nb  3  for  both  terms  taken  together  mufl  n  o  ; 
thus,  —  na  -f-  nb  m  o  :  Cosfequently,  —  X  —  gives 
+  for  the  produft.  ^.  £.  D. 


CHAP.     V. 

of    p  iris  IQ  ft. 

DIVISION  being  the  converfe  of  multipli 
cation  ;  it  follows,  that  the  quotient  muft  be 
iiich  a  quantity,  that  if  multiplied  with  the  divifor, 
will  produce  the  dividend  >  confequently,  like  figns 
in  divifion  give  -f,  and  unlike  figns  -—for  the  quo 
tient. 

CASE    I. 

Wktrn  the  dimjarls  aftmfle  quantify, 
R  U  L  E. 

T.  WRITE  down  the  quantities,  in  form  of  a  vul 
gar  fradtion,  having  the  divifor  for  the  denominator. 

2.  EXPUNGE  all  thole  quantities  in  the  dividend 
and  divifor,  that  are  alike  5  and  divide  the  co-effi 
cients 


dents  of  the  quantities  by  any  number  that  will  di 
vide  them  without  a  remainder  ;  the  refult  will  be 
the  quotient  fought. 

EXAMPLES. 


^u  the  quotient  >  ==i2j-2S  -=* 

2  a  22  a 


ii  adz  —  8  Jcz 


-  42 


IF  you  divide  any  quantity  by  itfelf,  the  quotient 
will  be  unity  or  i  :  Thus,  -mi  jfor  if  the  quotient  be 

multiplied  with  the  divifor,  the  produd  will  be  the 
dividend  ;  thus,  x  x  i  ~  x  -  Confequently,  if  any 
term  of  the  dividend  be  like  that  of  your  divifor, 
the  quotient  of  that  term  will  be  i  :  As  in 


"U 


3 
CASE     II. 

When  the  divifor  and  dividend  are  both  torn  found 
quantities. 

RULE. 

1.  RANGE  the  quantities  in  the  divifor  and  divi 
dend,  according  to  the  order  of  the  letters. 

2.  FIND  how  often  the  firft  term  of  the  divifor  is 
contained  in  the  firft  term  of  the  dividend,  and  place 
the  refult  in  the  quotient.  3. 


3.  MULTIPLY  the  quotient  term  thus  found,  with 
the  whole    divifor,   fubtrad  the  product  from  the 
dividend,  and  to  the  remainder  bring  down  the  next 
term  o&the  dividend  ;  which  forms  a  new  dividend* 

4.  DIVIDE  the   firft  term  of  your  new  dividend, 
by  the  firft  term  of  your  divifor,  as  before  ;   and  fo 
on,  until  nothing  remains,  as  in  common  Arithmetic, 
and  you  will  have  the  quotient  required. 

EXAMPLES. 

Suppofe  it  is  required  to  divide  i  yyy  +  %yy  -f-8^ 
\)j  yy  +  iy  •>  which  being  ranged  as  directed  in  the 
rule,  the  operation  will  ftand 

Thus,  yy  +  zy)iyyy  +  *yy  +  *y(*y  +4 

lyyy  +4  yy 


Here  thefirft  term  of  the  dividend,  which  is  zyyy, 
being  divided  by  tbejirfl  term  of  the  divifor  yy,  the  quo 
tient  is  iy\  which  being  placed  in  the  quotient  as  in 
vulgar  Arithmetic,  and  multiplied  with  all  the  terms  of 
the  divifor,  the  produft  is  iyyy  +4jv>',  whichjubtraft- 
cd from  the  dividend,  the  remainder  is  4jvy,  to  which 
annex  the  next  term  of  the  dividend  %y,  the  new  divi 
dend  becomes  ^yy  -fr-8j>,  and  dividing  q.yy  by  yy,  the 
quotient  is  4  ;  which  being  annexed  to  the  quotient  term 
before  found,  and  multiplied  with  every  term  of  the  di- 
vipr,  produces  ^yy-\-^y^  which  fubtrafted  from  the 
lafl  dividend,  the  remainder  is  nothing ;  and  having 
brought  down  all  the  terms  of  the  propofed  dividend , 
the  work  is  done ;  therefore,  iy  -{-4  is  the  true  quo 
tient,  for  iy  -f  4  X-^y  +  2j>  ~iyyy  +8jjy  4-8  JT— 
the  given  dividend.  Divide' 


Divide   6  avv  —  $av — -a^  4-2  ^ +2v— •  i  by 
v —  i. 

OPERATION. 


f\   sificr 

*        *  ~4vy 


1>        *    —I 

2V  — I 


Divide  vvv  —yyy  by  v  —  j^. 
OPERATION. 

v  —y)  vvv  >—yyy(vv+vy+yy 


*    +  vyy  —  yyy 
-f-  vvy  —  vyy 


—  yyy 

vyy—  yyy 


Divide  i  by  i«*»v 


OPERATION, 


256 


OPERATION. 


IN  this  example,  the  divifor  cannot  exactly  be  found 
in  the  dividend,  without  a  remainder  >  and  you  have 
what  is  called  an  infinite  (tries  for  the  quotient ;  that 
is,  if  the  divifion  could  be  carried  on  ad  iufinitum, 
you  would  have  a  feries  of  terms  for  the  quotient, 
that  would  come  infinitely  near  to  an  equality  with 
the  true  quotient,  and  therefore  might  be  confidered 
as  fuch  ;  for  when  ratios  from  that  of  equality,  are 
but  indefinitely  little,  or  lefs  than  can  be  afiigned, 
they  may  be  confidered  as  equal  -,  but  as  it  is  impof- 
fible  to  carry  on  the  divifion  ad  infini  titty,  or  take  in 
a  fufficient  number  of  terms  to  exprefs  the  true  quo 
tient  :  Therefore,  in  general  you  need  only  take  a 
few  of  the  leading  terms  for  the  quotient,  which  will 
be  fufficiently  near  for  mod  purpofes.  :  But  more  of 
this  in  its  proper  place,  fince  the  knowledge  of  Alge 
braic  fractions,  is  in  moft  cafes,  abfolutely  necefTary, 
in  order  to  obtain  an  infinite  feries  by  divifion. 

CASE      III. 

When  tie  quantities  in  the  dfoifor  cannot  be  found  in 
the  dividend. 

RULE, 


,     RULE. 

PLACE  the  dividend  above,  and  the  divifor  belovy 
a  fmall  line,,  in  form  of  a  vulgar  fraction  j  and  the 
expreffion  will  be  the  quotient  required. 

EXAMPLES. 

The  quotient  of  a  divided  by  b,  is  -, 
The  quotient  0/21  Ix  -~  d  ——7 — • 


The  quotient  cf%ac  +  dc-~zx  + 

^      8  <ic  +  dc 

zx  -|-  ab 

C  H  A  P.     VI. 

INVOLUTION   of   WHOLE     g>JJdN- 


INVOLUTION  is  the  raifing  of  powers 
from  quantities  called  roots,  and  differs  from 
multiplication  in  this,  viz.  that  in  involution  the 
multiplier  is  conftant,  or  the  fame  j  therefore  when 
any  quantity  is  drawn  into  itfelf,  and  afterwards  into 
that  product,  and  fo  on,  the  mode  of  operation  is 
called  involution,  and  the  number  produced,  the 
power,  whofe  height  is  ufually  denominated  by  plac 
ing  numeral  figures  over  the  right  hand  of  the  root> 
or  quantity  to  be  involved,  and  are  called  indices  or 
exponents  of  the  powers  which  they  denominate  : 
.Thus,  a*~aa  denominates  the  fquare  of  a,  a*~ 

K  k  aaa 


the  cube  of#,  a*-  the  fourth  power  of  a\  and 
generally,  an  the  n  power  of  a. 

INVOLUTION  of  firnple  quantities  is  performed  by 
the  following 

RULE. 

MULTIPLY  the  index  or  exponent  of  the  given 
quantity  or  root,  with  the  exponent  which  denomin 
ates  the  power  required,  making  the  produft  the 
exponent  of  the  power  fought. 

Note.  If  the  quantities  to  be  involved,  have  co-effi 
cients,  the  co-  efficients  muft  be  involved  as  in  vul~ 
gar  Arithmetic^  to  thejame  height  as  the  index  of 
the  fower  required  denotes. 

EXAMPLES. 


Jquare  of  n  ±=  a     •  ~  =  a*  j  the  cube  of  a=^ 

-a*i  the  cube 
•>  tic  tfb  fower  of 
:z:  256  xl*y*  5 
the  n  power  of  x~  xl^n=x>1. 

IF  the  quantity  propofed  to  be  involved  is  pofitive, 
all  its  powers  will  be  pofitive:  Alfo,  if  the  quantity 
propofed  be  negative,  all  its  powers  whofe  exponents 
are  even  numbers,  will  likewife  be  pofitive  ;  becaufe 
any  even  number  of  multiplications  of  a  negative 
quantity,  gives  a  pofitive  one  for  the  product,  fince 
—  X  —  gi^s  -f-5  confequently  —  X  —  X  —  X  —  = 
-f  X+  f°r  tne  produdl  ;  therefore,  that  power  of  the 
negative  quantity,  only  is  negative,  when  its  expo 

nent 


nent  is  an  odd  number :  As  may  be  feen  in  the  fol 
lowing  form, 


— •  a  the  root,  * 


—  a  the  roof 

—  a  3  ir  cube 

—  a  the  root 


a*—  tf    power 
—  —  a  the  root 


I  — a*-s=z$tb  power. 

INVOLUTION  of  compound  quantities,  is  perform 
ed  by  the  following 

RULE. 

MULTIPLY  the  root  into  itfelf,  and  then  into  that 
product,  and  fo  on,  until  the  number  of  multiplica 
tions  are  one  lefs  than  the  exponent  of  the  power  re 
quired  ;  the  refult  will  be  the  power  fought, 

EXAMPLES, 

L,et  the  binomial  <?-f-£be  involved  to  the  5th 
power- 


OPERATION, 


(      260      ) 

,  _  -|      -  •     •   - 

OPERATION. 


a+b  the  root 
a  +  b 

aa-\-ab 


aa  ~f-  lab  -f-  bb  — 


aaa~\-i.aab-\~abb 
-\-aab  •\-<2. 


aaab  +  ^aabb-\-^abbb  -\-  bbbb 


aaaa  -\-  ^.aaab  +  6aabb  +  ^abbb  +  bbbbit^th  fower 


aaabb  +  ^aabbb  +  ^^^^^ 
4-  aaaab-\-^aaabb  -\>  6aabbb  +  A^abbbb  -f-  bbbbb 


Involve 


Involve  a  —  b  to  the  3d  power. 
OPERATION. 


z  — lab +b*~  id  power 
a  —  b 


* — &3~  3d  power. 

IT  is  to  be  obferved  in  the  foregoing  examples. 

1 .  THAT  all  the  terms  in  the  feveral  powers,  railed 
from  the  binomial  a-\-by  are  affirmative. 

2.  THE  terms  in  the  feveral  powers  raifed  from  the 
refidual  a  —  by  have  the  figns  -f-  and  — ,  alternate 
ly  ;  the  firfl  term  being  a  pure  power  of  a,  is  confe- 
quently  affirmative ;  the  fecond  term  hath  a  nega 
tive  fign,  and  fo  on,  alternately  j  but  b  is  no  where 
found  negative,  only  where  its  exponent  is  an  odd 
number;  as  in  a2  —  ^a^b+^ab* — b*  ;  where  the 
fecond  and  fourth  terms  are  negative,  becaufe  the  ex 
ponent  of  b  in  thoje  terms,  is  an  odd  number. 

3.  THAT  the  firft  term  of  any  power,  either  of  the 
binomial  or  refidual,    hath  the  exponent  of  the  pow 
er  :  That  is,  the  index  of  the  firft  term,  is  equal  to 
the  index  of  the  power  ;  bui  in  the  reft  of  the  terms 
following,  the  exponents  of  the  leading  quantity,  de- 
creafe  in  arithmetical  progreflion,  unity  or  i,   being 
the  common  difference  ;  fo  that  the  quantity  a  is 

never 


(        262        ) 

never  found  in  the  Lift  term  j  but  the  exponents  of 
by  on  the  contrary,  increafe  in  the  fame  progreflion 
that  the  exponents  of  a  decreafe  ;  that  is,  the  quan 
tity  b>  is  not  to  be  found  in  the  firfttermj  but  in  the 
fecond  term,  its  exponent  is  unity  or  i  5  in  the  third 
term  2,  and  fo  on  in  the  faid  arithmetical  progref- 
fion,  to  the  laft  term,  where  its  exponent  is  equal  to 
the  exponent  of  the  power. 

4.  That  the  number  of  terms  in  any  power,  is  one 
more  than  the  number  which  denominates  that; 
power. 

HENCE  from  the  foregoing  obfervations  it  follows. 

i.  THAT  the  fum  of  the  exponents  of  both  quan 
tities  in  any  term,  are  equal  to  the  exponent  of  the 
power  in  which  thofe  terms  belong  :  Thus,  the  6th 
power  of  a  +b  zr  a6  -\-6a*b  +  i$a*b*  +  2oa*l>3 
*±  i$a*&*-\-6at>5  +b6y  where  you  will  pleafe  to  ob- 
ferve,  that  the  fum  of  the  exponents  of  a  and  by  in 
any  term,  are  equal  to  the  exponent  of  the  power  : 
Thus  in  the  third  term,  the  exponents  of  a  and  by 
are  4  and  2,  whofe  fum  —  6 ^exponent  of  the  pow 
er. 

•2.  THE  method  of  writing  without  a  continual  in^ 
volution,  the  terms  in  any  power  of  a  binomial,  or 
refidual  quantity,  without  their  co-efficients  :  Thus 
the  terms  of  the  4th  power  of  x +y  without  their 
co-efficients,  will  ftand  thus  :  x^+x^y+x^y*  *%-xy* 
and  the  terms  of  the  4th  power  of  #— y~x* 
,?* — xy 3  -f-j*. 

IN  order  to  find  the  co-efficients  ofthefeveral 
terms,  it  is  necefTary  to  have  the  co-efficient  of  one 
of  the  terms  given  :  And  becaufe  the  firft  term  or 
leading  quantity  is  a  pure  power,  having  its  index 
equal  to  the  index  of  the  given  power  ;  its  co-effi 
cient  is  therefore  unity  or  i  :  Confequently,  you 

have 


have  the  co-efficient  of  the  firft  term  given  ;  thence 
to  find  the  co-efficients  of  the  reft  of  the  terms  by 
the  following 

RULE. 

DIVIDE  the  co-efficient  of  the  preceding  term,  by 
the  exponent  of  y  in  the  given  term  ;   the  quotient 
multiplied  with  the  exponent  of  x,  in  the  fame  term, 
iiicrcafed  by  x,  will  give  the  co-efficient  required. 
Or, 

MULTIPLY  the  co-efficient  of  any  term,  with  the 
exponent  of  the  leading  quantity,  in  the  fame  term  ; 
the  product  divided  by  the  number  of  terms  to  that 
place,  will  give  the  co-efficient  of  the  next  fubfe- 
quent  term. 

EXAMPLES. 


Given  #4+#3^4-#^7+*y3+.74j  to  find  the  co-ef 
ficients  of  the  feveral  terms. 

Firft,  the  co-efficient  of  #4  is  i  ;  thence  to  find  the 
co-efficient  of#3j  :  And  becaufe  the  exponent  ofy 

in  the  given  term,  is  unity  or   i  ;  then  per  rule,   I 
=  iX4=4>  the  co-efficient  required  :  Again, 

i=rlX3n—  =6,  the  co-efficientof  the  third 
2  22 

term  ;  and  -Xi+  izr-X2—  —  ~4>  the  co-efficient 

3  33 

of  the  fourth  term  ;  but  the  next  term  hath  the  ex 
ponent  of  the  power,being  the  lad  term  of  the  4th  pow 
er  of  A'-f-jy,  and  confequently,  its  co-efficient  an  unit 
or  i  .  Therefore,the  co-efficients  of  the  feveral  terms 
of  the  4th  power  of  x+yt  are  i,  4,  6,  4,  i. 

HJSNCIZ 


HENCE  you  may  obferve,  that  the  co-efficients  of 
the  feveral  terms  increafe,  until  the  exponents  of  x  and 
y  become  equal  to  each  other,  and  then  decreafe  in 
the  fame  order  in  which  they  increafed.  And  geni 
ally,  the  co-  efficients  of  the  terms  increafe,  until  the 
exponents  of  the  two  quantities  become  equal  in  one 
term,  if  the  exponent  of  the  power  is  an,  even  num 
ber  ;  arid  when  the  exponent  is  odd,  two  of  the  terms 
will  have  equal  co-efficients,  and  then  clecreafe  in 
the  fame  order.  Therefore,  in  finding  the  co-effi 
cients,  you  need  only  obtain  the  co-efficients,  until 
they  decreafe  ;  the  reft  of  the  terms  having  the  fame 
co-efficients  decreafmg. 

THE    n    power  of    ^  +  «iif+»^""V-f-»X 

-z^-v+ix^x^*'-^'  &c.  to- 

2  23 

i,  terms. 


Let  a  -}-  b  4-  c  be  involved  to  the  fecond  power, 
OPERATION. 


1  -f  ab  4-  ac 
+l* 
+  ca 


a*  -J-  iab  4-  iac-{-  b*  -fa  bc+  c*^  id  power. 


CHAP.     VII. 

Of  MULTIPLICATION  and  DIVISION 
of  POWERS  of  the  fame  ROOT. 


M 


ULTI  PLICATION  of  powers  of  the 
fame  root,  is  performed  by  the  following 

RULE. 


ADD  the  exponents  of  the  powers  together,  and 
make  their  fum  the  exponent  of  the  product. 

EXAMPLES. 

=:  6s  rz 


7776  ;  6  #3  X4  #4=: 

a6~  —  a10  ;  alfoy  —  0'X  —  ^4  =  ^3  i  in  like  man 


5   0m/ 

unmerfally> 


DIVISION  of  powers  that  have  the  fame  root,  is 
effected  by  the  following 

RULE. 

wfl 

FROM  the  exponent  of  the  dividend,  fubtradt  the 
exponent  of  the  divifor,  and  the  remainder  will  be 
the  exponent  of  the  quotient. 

LI  EXAMPLES. 


(      a66      ) 

EXAMPLES. 


4— .2_<253Z 

0  —  5 


HENCE  it  follows,  that  in  divifion  of  powers  which 
have  the  fame  root,  if  you  divide  a  lefs  power  by  a 
greater,  the  exponent  of  the  quotient  will  be  nega 
tive  ;  for  we  have  fbewn,  that  to  divide  any  power 
of  a  by  #,  is  to  fubtract  one  from  the  exponent  of  the 

power  of  a:  Thus,    -z:^1;  therefore,     -~a 
a  a 

~a°  5  but  ~i  by  the  nature  of  divifion  j  confe- 
a 

quently,  #°  —  i  by  equality  j    and  therefore,    Izif- 

a      a 

o — i — i^       j  i  #° o— 2 —2  ^        , 

a*~~aT~ 

ib    on    for    any    power   of  ~;  Likcwife,  ^  I  az= 
x-\-y\  "     ~^Hhjl  =  (becaufe,  '  — 13)      i  '; 

confequently,    .  l      ^^!J2l  —  .y^,y|""     .  therefore. 


' 


J  ;     And  generally, ' 
~x+j\~~n.   Therefore,  0°,  4     I>  a     2,  ^     3^    anj^ 


*       a*       a* 

and  of  which  they  are  pofitive  pow- 


|      x+y 
crs. 

HENCE  the  propriety  of  ufmg  negative  exponents. 

THE  multiplication,  and  divifion  of  powers  which 
have  the  fame  root,  having  negative  exponents,  is 
performed  by  the  fame  rule  as  thofe  powers  which 
have  affirmative  ones  ;  that  is,  add  the  exponents  of 
the  factors  in  multiplication,  and  in  divifion  fubtracT: 
them. 

EXAMPLES. 

»™»*>O  iHi«*»/t  MMW  O   wmvmm  A  •"•'•^O 

A        multiplied  with  a    ^  r=  a  *  =  a       ; 

a~3  X  a~~l  =  a~~l~3=ia'~'*  •>  a~*  X  a*    ~ 

*\/a  X 


-r-0     ^~   (by     the    nature    of    fubtra&ion) 
==tf-3==I^3  .  and  ,f-3  -f^-6  zz^-3 

5  but  by  the  nature  of  multiplication  and 
divifion,     a~~^~d~'  ~a     ^-r-^"*""1^  X  #      ^^ 

=  a3-,    likewife, 


(      263      ) 


CHAP,     VIII. 

EVOLUTION    of    WHOLE 

TIES. 

EV  O  L  U  T  I  O  N  is  the  unfolding  of  powers 
produced  by  involution  3   thereby  difcovcriog 
-the  roots  with  which  they  are  compofed,  and  is  there 
fore  the  reverfe  of  involution, 

THE  rule  for  evolution  of  powers,  whofe  roots  are 
fimple  quantities,  flows  from  this  confideration  ;  that 
to  involve  any  fimple  quantity  to  any  power,  is  to 
multiply  the  exponent  of  the  quantity,  with  the  expo 
nent  of  the  power ;  making  the  product  the  exponent 
of  the  required  power;  consequently,  if  the  expo 
nent  of  the  power,  be  divided  by  the  index  which 
denominates  the  root  required,  the  quotient  will  be 
the  exponent  of  the  root.  Therefore,  when  the  ex 
ponent  of  the  power  whofe  root  is  required,  is  not 
a  multiple  of  the  number  which  denominates  the 
kind  of  root  required  -,  it  follows,  that  the  root  will 
be  exprefled  by  a  fractional  exponent :  Thus,  the 

s  £. 

fquare  root  of  a5  —  a*,  and  the  cube  root  of  ^4=^T. 
Whence,  we  have  the  following  rule  for  evolution  of 
fimple  quantities. 

RULE, 

EXTRACT  the  root  of  the  co-efficient,  as  in  vul 
gar  arithmetic,  and  divide  the  exponent  of  the  power, 


fay  the  index  of  the  root  rquired  \  making  the  root 
of  the  co-efficient,  the  co-efficient  of  the  root. 

EXAMPLES. 


cube  root  of  a9—ar~  a*  :  The  fquare  root  of 
i2a*~  2  a*  :  <?be  cube  root  of  64.x9  a3—3  ^64 
4*3a  :  fhe  tfb  root  of  256  a*b**  =1*^/256 

4.  I 

X  <?T£T  ~  4  #£3  :  ^be  cube  root  of  —  27  #3zi  —  3/2  J. 
But  the  fquare  root  of  a  negative  quantity  ;  as  —  #% 
cannot  be  afTigned,  becaufe  no  even  number  of  mul 
tiplications,  either  of  a  pofitive  or  negative  quantity, 
can  give  a  negative  one  for  the  produd,  as  was  fully- 
explained  in  chapter  vi  5  therefore,  the  fquare  root 
of  —  #*  is  an  imaginary  quanritiy  :  And  fince  the 
iquare  of  any  negative  or  pofitive  quantity,  is  always 
pofkivej  it  follows,  that  the  fqjare  root  of  x*  may 
be-M,  or  —  x.  Therefore,  when  the  number  which 
denominates  the  root  to  be  extracted,  is  odd,  thefign 
of  the  root  will  the  fame  as  the  fign  of  the  power  ; 
and  when  the  number  which  denominates  the  root, 
is  even,  the  fign  of  the  root  may  be  either  -j-  or  —  : 
Thus,  the  cube  root  of  —  2jat5b9~  —  3  asb3, 
and  the  4th  root  of  16  £*#*—  2  a*x  or  —  2  a*  '  x  ;  the 

the  ;;  power   of  .vwzr^-; 

EVOLUTION  of  compound  quantities,  requires  a 
different  method  of  proceeding  from  that  of  fimple 
ones. 

To  extract  the  fquare  root  of  a  compound  quan 
tity  we  have  the  following 

RULE. 


RULE. 

i.  RANGE  the  quantities  according  to  the  order 
of  the  letters,  fo  that  the  firft  term  fhall  have  the 
index  of  the  power. 

.    2.  FIND  the  root  of  the  firft  term,  as  in  evolution 
of  fimple  quantities,  and  place  it  in  t}he  quotient* 

3.  SUBTRACT  the  fquare  of  the  root  thus  found, 
from  the  firft  term  of  the  power  propofed,  and  to  the 
remainder  bring  down  the  reft  of  the  terms  for  a  di 
vidend. 

4.  DIVIDE  the  firft  term  of  the  dividend,  by  dou 
ble  the  root,  and  write  the  refult  in  the  quotient, 
for  the  fecond  term  of  the  root, 

5.  ADD  the  laft  term  of  the  quotient  to  your  divi- 
for,  and  multiply  their  fum  with  the  faid  quotient 
term,  fubtracling  the  product  from   the  dividend  ; 
and  fo  on,  to  obtain  the  next  term  of  the  root,  by 
the  help  of  jthofe  already  found,  in  the  fame  manner 
as  the  fecond  term  was  obtained  by  the  help  of  the 
firft. 

EXAMPLE. 
I  Extract  the  fquare  root  of  a%  +  2  ay  +  y '  ~f-  2  za 

-f  2J2  4-  Z\ 

The  fquare  roof  of  the  fir fl  term  viz.  #  %  is  ay  which 
being  placed  In  the  quotient,  is  the  firft  term  of  the  roof, 
(fee  the  operation  annexed)  which fquared  and  fubtraffed 
from  the  firft  term  of  the  propofed  power,  leaves  no  re 
mainder  ;  the  reft  of  the  terms  being  brought  down  for 
a  dividend,  the  firft  term,  viz.  i  ay  divided  by  la 
(the  double  of  the  root)  gives  y  for  the  fecond  term  of 
the  root ;  which  with  the  divifor,  being  multiplied  with 
y>  and  the  produtt  fubtrafted  from  the  firft  terms  of  the 

dividend, 


dividend,  the  remainder  is  nothing;  the  remaining 
terms  being  brought  down  as  before  and  divided  by  the 
double  oftbe  twofrjt  terms  of  the  root,  gives  zfcr  tlse 
third  term  of  the  root,  which  added  to  the  divifor  and 
multiplied  with  z,  the  frodutt  fubtrafted  as  before, 
leaves  no  remainder  :  Therefore,  the  root  fought,  is 


a 


OPERATION. 


And  univerfallyy  to  extraEl  any  root. 

R  ULE: 

1.  RANGE  the  terms  of  the  given  power,  as  in  the 
laft  rule. 

2.  EXTRACT  the  root  of  the  firft  term  as  before, 
and  place  it  in  the  quotient  for  the  firft  term  of 
the  root. 

3.  SUBTRACT  the  power  of  the  root  thus  found, 
and  to  the  remainder  bring  down  the  next  term  for 
a  dividend. 

4.  INVOLVE  the  root  to  a  dirnenfion  lower  by  unity 
than  the  number  which  denominates  the  root  requir 
ed,  and  multiply  the  refult  with  the  index  of  the 

root 


root  to  be  extracted,  which  product  call  your  divi- 
for. 

5.  FIND  how  often  the  divifor  is  contained  in  the 
dividend,  and  write  the  refult  in  the  quotient  for  the 
fecond  term  of  the  root. 

6.  INVOLVE  the  whole  of  the  root  thus  found,  tc 
the  dimenfion  of  the  given  power,  and  fubtra<5t  the 
refult  from  the  given  power  \  and  call  the  remainder 
a  new  dividend, 

7.  INVOLVE  the  whole  of  the  root  in  the  fame  man 
ner  as  you  did  the  firfl  term,  and  multiply  the  refult 
as  before  for  a  new  divifor. 

8.  DIVIDE  as  before,  and  the  refult  will  be  the 
third  term  of  the  root;  and  fo  on,  till  the  whole  be 
finifhed. 

EXAMPLES. 

Required  the   fquare  rootofi6jp6 — 

+96^+64. 

OPERATION. 


— - 8  is  the  roof  required. 

Required 


(      273      ) 

Required  the  cube  root  of  8«3-J- 
. 

OPERATION. 


#        *  *        * 

y  ia+  b,  is  the  root  required. 


CHAP.      IX. 

Of    ALGEBRAIC    FRACTIONS   or 
BROKEN    QUANTITIES. 

ALGEBRAIC  fra&ions  are  formed  by  the 
divifion  of  quantities  incommenfurable  to  each 
other  :  Thus,  if  x  is  to  be  divided  byjy,  it  will  be 

X 

(by  cafe  in,  of  algebraic  divifion)  -,  which  is  an 

y 

algebraic  fra&ionj  wherein  x  is  the  numerator  and_y 
the  denominator.  When  fractions  are  connected 

v  /*v  -jfj » ^ 

with  undivided  quantities  ,  as  a  -+--.  and  a  +          . 

y  a+b 

they  are  called  mixed  quantities  j  alfo,  if  the  denom 
inator  is  lefs  than  the  numerator,  the  fraction  is 
called  improper. 

THE  various  operations,  neceflary  in  managing 
algebraic  fractions,  arc  comprifed  in  the  following 
problems. 

Mm  PROB. 


PROBLEM    I. 

3"0  reduce  a  mixed  quantity  to  an  improper  fraRion 
of  equal  value. 

RULE. 

MULTIPLY  the  denominator  of  the  fraction  with 
the  integral  part,  to  which  product  add  the  numera 
tor,  and  under  their  fum,  fubfcribe  the  denominator, 
for  the  fraction  required. 

EXAMPLES. 

•    tf^li-rr- 


^  —  2 

:  —  iv  — 


a  —  2  ^  —  2 

PROBLEM      II. 

^i?  reduce  an  improper  f  ration  to  a  whole  or  mixed 
quantify. 

RULE. 

DIVIDE  the  numerator  by  the  denominator  for  the 
integral  part,  and  write  the  denominator  under  the 
remainder  for  the  fractional  part  ;  and  you  will  have 
the  number  required, 

EXAMPLES. 


(      27S      ) 

EXAMPLES. 


a  —  b 
PROBLEM     III. 

TV  reduce  fractions  of  different  denomination  s>  to 
fractions  of  the  fame  value  ,  that  Jhall  have  a  common 
denominator* 

RULE. 

1.  REDUCE  all  mixed  quantities  to  improper  frac 
tions. 

2,  MULTIPLY  every  numerator  feparately  taken, 
into  all  the  denominators  except  its  own,  for  the 
feveral  numerators,  and  all  the  denominators  toge 
ther  for  the  common   denominator,    which   being 
wrote  under  the  feveral  numerators,  will  givti,  the 
fractions  required. 

EXAMPLES. 

Reduce   1  and  -L  to  fractions  of  the  fame  value, 

2         4 

.having  a  common  denominator.     Firft,  ArX4^4-v 
and  jX2~2j  for  the  numerators  :  Then,  2X4=8, 


the  common  denominator.     Therefore,    il  and  22 

o  b 

are  the  fractions  required. 

Reduce  -,   5,   and  -    to  equivalent  fractions, 
y    v  c 

having  a  common  denominator, 


s=w*'1 
~cyz  V 
~ayv  J 


—numerators. 


izrvyz:  common  denominator. 
Therefore,   fZl,   2iand  22  are  the  fractions  re- 


quired;  which  are  rjefpeftively  equal  to  -,   -,   ?. 

for       -=  (by  the  nature  of  divifion)- $  and  the  like 
for  the  reft.    Whence,  &c. 

Reduce,    -' ,    — ,   and  2!  to  a  common  de 
nominator,  retaining  their  refpective  values. 

"lav — iv*  1 

;=2i> 3  b      |>  =  numerators. 

zv  X  2  X  ^^14^*  —common  denominator. 

,_,,          ..  2^<y~~~*2'ya       2*y3^  i    A^^V  i 

i^ierefore,  , . — ,  — r,   and  2-^,  are  the 

fra<5lions  required. 

_,    H,   and  — £  reduced  to  a  common  denomin- 
x     ba  ax 


x*a*x      cax*  ,  b*acx 

ator,  are  -  '-  -  ,  .    and  7-- 


PROBLEM    IV. 

^  greateft  common  meajure  of  algebraic 
fractions. 

RULE. 


R  U  L  E. 

1.  RANGE  the  quantities  as  in  divifion. 

2.  DIVIDE  the  greater  quantity  by  the  lefs,  and 
the  laft  divifor  by  the  laft  remainder,  until  nothing 
remains;   taking  care  to  expunge  thofe  quantities 
that  are  common  to  each  divifor ;  and  the  laft  divifor 
will  be  the  greateft  common  meafure  required. 

EXAMPLES. 

~ 


Find  the  greateft  common  meafure  of 
OPERATION. 


Therefore,  v  —  a,  is  the  greateft  common  meafure 
required. 

Z  7fc 

Find  the  greateft  common  meafure  of- 

* 


OPERATION. 


OPERATION, 
a*  —  »  at>  +  l>*)  a*  ~  !>*  (i 


O, ( by  cafting  out  ib)  a — b}  a* — iab+  &* 

a1 —  ab 


^Therefore,  a—b,  is  the  great  eft  common  meafure  r^ 


PROBLEM    V. 

To  reduce  fractions  to  their  leafl  terms* 
RULE. 

1.  FIND  their  greateft  common  meafure  by  the 
laft  problem. 

2.  DIVIDE  both  terms  of  the  propofed  fra6lion  by 
their  greateft  common  meafure,  and  the  quotients 
will  be  the  refpedivc  terms  of  the  fraftion,  reduced 
to  its  leaft  terms. 


EXAMPLES. 


EXAMPLES. 

Reduce   *a  +  a to  its  leaft  terms. 

xy*  +  y*a 

Firft,  xa  +  a*)  xy*  -\-y*a 
Or,  x+a)xy*+y*a(y* 


*          # 

*Theny  x  +  a)  xa  -{•  a%  (^=  numerator. 
xa  +  a* 


a)  xy*  +  y^a  (j>*  rr  denotftinator. 


Therefore,  —  is  tbe  propofed  fraftion  in  its  haft 

terms. 

Reduce  -Jl—fl  —  to  its  leaft  terms. 
y  s  —  x  2  y  3 

Firft)  the  great  eft  common  meafure  is  y*—  x*  : 


then, 


PROBLEM    VI. 

jTi?  add  algebraic  fractions* 
RULE. 

i.    PREPARE  the  given  fra&ions  by  redu<5lion  i 
that  is,  mixed  quantities  muft  be  reduced  to  improp 


cr  fractions,  and  all  fra&ions  to  a  common  denom 
inator. 

2.  ADD  all  the  numerators  together,  under  which 
write  the  common  denominator;  and  you  will  have 
the  fum  required. 

FOR,  put  -z=<z,  and  -.—  b  $  then  will  v  ~ya  and 

y  y 

z~yb  by  the  nature  of  divifion;  confequently  ya  + 
j£zz  v-\-z,  and  therefore  by  divifion  a+b  zz 


But,  a  +  b^+*->  confequently,   2+;=!±2  ; 

jr   ^  y    y     y 

which  is  the  fame  as  the  rule. 


EXAMPLES. 


Given  -,  ^  and  1^  to  find  their  fum. 
66          6 


u  -\-u  -j-4z  ^  2.  u  +  4  2  tf»^/  —  i!_r:  /^^  requir. 

6 

Having  ~,   —  and^  given  to  find  their  fum. 
0  2      y          u 

Firft,  uxyX  u  ^  a*Jj  and 


-f-  6  j  •--  2  #j  z=^fum  required. 

1^+  ^  -K2*—4*-!  S^— 
"~      ' 


- 
3  "~2^  '   3  6  a 

PROBLEM      VII. 

Tojubtraff  one  /ration  from  another. 

RULE. 
i,  PREPARE  the  quantities  as  in  the.iaft  problem 

2. 


2.  SUBTRACT  the  numerator  of  the  fubtrahend 
from  the  numerator  of  the  other  fra&ion,  and  write 
the  common  denominator  under  their  difference  ; 
and  you  will  have  the  fra&ion  required. 

1)  CL 

FOR  put  -=  m  and  -  —  n  $  then  v—ym  and  a  zr 

y  y 

yn  ;  alfo,  yn  —ym  zzza  —  v  by  equality  ;  and  di 
viding  the  whole  by  j,  it  will  ben  —  m—  *""??.•,  but 

y 

the  difference  of  m  and  #,  is  manifcftly  equal  to  the 
difference  of  2  and  -5  confequently,  !—  -=.£lir, 

jr      7  y  y     y 

Hence,  &c. 


From  f  take—.    Firjt,  aXal=a*l>,  and  c^b 
b         ab 


~cbx>  alfo>  bKab<=,ab\    ^Therefore, 

ae* 

*re  the  fractions  reduced-,  and  -    —  =  difference 


c*—x%  c*-}-^1* 

required.    From  -   take  '-,  and  it 

7" 

b 


—  ;£.  The  f  rations  reduced  arc 
4  8 

/on?,  —4^4-6^,,^;,,,  ^r^^  required,  by  the  na~ 

O  9 

ture  offubtratliw. 

Nn  PROB, 


(        282        )    V 

PROBLEM    VIII. 

fo  multiply  fractional  quantities  together. 
RULE. 

MULTIPLY  ttie  numerators  together  for  the  nume 
rator  of  the  product,  and  the  denominators  together 
for  the  denominator  of  the  produd:  5  and  you  will 
have  the  product  required. 

fTj  +  j* 

FOR  put-zr  m  and  -~  n\   then  v  zi  zm  and  a  ~ 
z  b 

In  j  alfo,  bnx%m~  a  X  v  >  that  is,  bznm  zz  ai)y  and 

dividing  \yftz,  nm  ~  —  5   but  m  X  n  ^:—  X-J 
bz  z    b 


fequently,       x-=—  :  Therefore, 
z     b     bz 

EXAMPLES. 


- 


3       4  3  4      3        12         4 

or,  ^y. 

PROBLEM     IX. 

To  divide  one  fraflion  by  another, 
RULE. 

MULTIPLY  the  denominator  of  the  divifor,.  with 
the  numerator  of  the  dividend,  for  the  numerator  of 

the 


,w-  (      383      ) 

the  required  quotient,  and  the  numerator  of  the  di- 
vifor,  with  the  denominator  of  the  dividend,  for  the 
denominator  of  the  quotient.  Or, 

INVERT  the  terms  of  the  divifor,  and  proceed  as 
in  multiplication. 

For  put  —  ~m  and  -  —  #  >    then  x~ym  and  z 
y  d 


"zzdn.  Multiply  z=dn  by  y>  and  it  will 

in  like  manner,    dx^dym-,    therefore,  •L—^-L.j 

aym     dx 

but  3  ?  z:  fby  divifion)  ~,    and  therefore  by  refti- 
ydm  m 

tution  ?-4-l=^  :  Cpnfequently,  &c. 
d    y     dx 

EXAMPLES, 

a     c    a^d    ad     ~      a     c     d^a     ad      ,    c 
T^----9->~~z.    Or,  7~:}z:-XI^:—  as  before  j 
b     d     cXd     cb  b     d     c     b      cb 

a—u  .  a-^-u  __a  —  u  X  v^va—uv  __  a  —  u  .     rp, 


fore,  in  divifion  of  fractions  that  have  the  fame  de 
nominator,  caft  off  the  denominators,  and  divide  the 
numerator  of  the  dividend,  by  the  numerator  of  the 
divifor,  for  the  quotient. 


Thus,  l£  ^Slxfe= 


3 

ay  ^6  a 


PROBLEM    X, 

fo  find  the  fowtrs  of  fractional  quantities. 

RULE, 


RULE. 


i.  PREPARE  the  given  fradHon,  if  need  be,  by  the 
rules  of  reduction. 

a.  INVOLVE  the  numerator  to  the  height  of  the 
power  propofed,  as  in  involution  of  whole  quantities, 
for  the  numerator  of  the  power  required. 

3.  INVOLVE  the  denominator  in  like  manner,  for 
the  denominator  of  the  aforefaid  power. 


EXAMPLES. 

Find 


X 


z=z  power  required. 


a*,  f^ 

uhe  4tb  power  of  —  •=.• 
zy 

PROBLEM    XL 

To  find  the  roots  of  fractional  quantities, 
RULE. 

1.  EXTRACT  th«  root  of  the  numerator,  by  the 
rules  for  extradting  the  roots  of  whole  quantities, 
for  the  numerator  of  the  root  required. 

2.  EATRACT  the  root  of  the  denominator  in  like 
manner,  for  the  denominator  of  the  required  root. 

'  EXAMPLES. 


a6 
Find  the  fquare  root  of  —  . 

* 


0%  /or  /£*  numerator  of  the  root, 
find  x  '  ''1~x*  for  the  denominator  of  the  root-,  there 
fore,  —  is  the  root  require^.  The  cube  root  of  ..  a  .. 

X*  Z3y* 

-JL    4/flfl-f*! 

""•zp"        2V6~~~z<;3' 

The  fquare  root  of  **~4*  +  4=flZ*. 
J*+6j  +  9     jK  +  3 

But  if  th?  propofed  quantity  hath  not  a  true  root 
of  the  kind  required,  it  muft  be  diflinguifhed  by  the 

fign  of  the  root:  Thus,  the  fquare  root  of?  ~~  X 

* 


or 


a* 


CHAP.     X, 

CONCERNING      SURDS      or     IRRA- 
TIO  NAL 


IF  the  whole  doftrine  of  furds,  with  every  thing 
therein,  which  might  be  of  ufe,  were  to  be  ex 
plained  according  to  the  methods  ufed  by  fome 
writers  on  the  fubje6b,it  would  become  very  complex, 
and  by  far  the  moil  intricate  and  dfficult  part  of  all 
Algebra  3  and  neceflarily  fwell  this  volume  beyond 

its 


its  defigned  limit  :  And  befides,  there  are  many 
things  in  the  explanation  and  management  of  furd 
quantities,  as  was  taught  by  many  writers  on  Alge 
bra,  which  were  then  thought  neceffary,  are  now  at 
moft,  confidered  as  ufeful.  We  fhall  therefore, 
endevour  on  the  one  hand-,  to  avoid  all  fuch  tedious 
redu&ions,  and  complicated  explanations,  as  would 
ferve  rather  to  puzzle,  than  inftruct  the  learner  : 
And  on  the  other  hand,  not  to  omit  any  thing  which 
is  neceffary,  either  in  the  explanation  or  management 
of  fuch  furds  as  generally  arife  in  algebraic  operations. 
A  SURD  quantity  is  that  which  has  no  exact  root  : 
Thus,  the  fquare  root  of  5  cannot  exactly  be  found 

in  fiinite  terms,  but  is  exprefied  by  5*,  or  \/5  $  the 

JL  —  .-£. 

cube  root  of  a  by  a3,  or   ay  ~  3  \/a  :  The  recipro 
cal  of  the  fquare  root  of  a-+y,  or  i  divided  by  the 

fquare  root  of  a  +y,  is  exprefied  by 


THEREFORE,  the  roots  or  irrational  or  furd  quan 
tities,  may  be  confidered  as  powers  having  fractional 
exponents;  that  is,  the  index  fhewing  the  height  of 
the  power,  is  here  placed  as  the  numerator  of  a  frac 
tion,  whofe  denominator  is  the  radical  fign. 

SECT.        I. 

Of    REDUCTION    of   SURD     QfTAN- 
<T  IVIES, 

REDUCTION  of  furds  has  the  following  problems. 

P  R  O  Bs 


PROBLEM    I. 

fo  reduce  a  rational  quantity  to  the  form  of  an  irra 
tional,  or  fur  d  quantity* 

RULE, 

INVOLVE  the  rational  quantity  to  the  height  of  the 
propofed  radical  fign,  or  index  fhewing  the  root  to 
be  extradled  ;  the  power  diftinguifhed  by  the  radical 
fign,  will  be  the  form  required. 

EXAMPLES. 


0,    reduced  to  the  form  of  the  3  i/x,  is  3  \/al  *3  rz 
3  y'tf3  ;  6  reduced  to  tie  form  of  the  fquare  root  of  2, 

lX2 


Reduce    2.  to  the  form  of  a  cube  root.     ^ 
4  4) 

zr~.3=3  y^^=fsrm  required  ;  u+y  reduced  to  the 

form  of  a  fourth  root,  is  *  \/  fu+y1^  '  =  4  \/  'u+y 

.   Alfoy  u~\/ii* 


5 


PROBLEM      II. 

fo  reduce  Jurds  of  different  radical  ftgns  to  the  fame* 
RULE. 

REDUCE  the  indices  of  the  furds  to  a  common  de 
nominator,  and  the  Curds  will  have  the  fame  radical 
fign  as  required,  EXAMPLES, 


(      288      ) 

EXAMPLES. 
The  3  \/  of  a,  and  the  \/  ofb,  reduced  to  the  fame 

1X2  1X3  2-  3 

I  t 

and-*\/b*.    2T  and  y*  reduced  to  the  fame 
1X3  2X4          s 


and  iz 

3X2  1X2 


3  x 

reduced  to  a  common  radical 


Jign>  are  ~£ 

i/^l1  :     Alfo,  y~  '  and  y~^~ 
PROBLEM    III. 

To  reduce  fur  ds  to  their  moftjimpk  terms. 
R  ULE. 

x.  DIVIDE  the  quantity  under  the  radical  fign,  by 
fuch  a  rational  divifor,  as  will  quote  the  greateft  ra 
tional  power  contained  in  the  propofed  furd  without 
a  remainder. 

2.  EXTRACT  the  root  of  the  rational  power,  and 
place  it  before  the  furd,  with  the  fign  of  multiplica 
tion,  and  the  propofed  furd  will  be  in  its  mod  fimplc  $ 
terms. 

EXAMPLES. 

Reduce  -/J?  to  its  mod  fmnple  terms. 

tiers 


(      289      ) 

~  16  the  greateft  rational  power  contained  in 
" 
;  therefore>the  /32  = 


4f 
3 

SECT.      II. . 

O/     ADDITION     of    SURD      $JJAN- 
ffTIES. 

ADDITION  of  furd  or  irrational  quantities^  confifts 
of  the  following  cafes. 

CASE      L 

When  the  propofedjurds  are  of  the  fame  irrational 
quantity  (or  can  be  made  Jo  by  reduftitn)  and  the  ra~ 
dlcal  fign  thejame  in  all. 

RULE. 

ADD  the  rational  to  the  rational,  and  to  their  funa 
annex  the  irrational  part  with  its  radical  fign. 

EXAMPLES. 
3/20+6/20  z:  3+6X  ^20—9^20 ;  \/ Z^K-SC, 


'1 

O  o  V"*- 


CASE    II. 

the  irrational  orjurd  quantity,  and  the  radi~ 
calftgn  are  not  tbefame  in  all. 

RULE. 

CONNECT  the  furds  with  their  proper  figns  +  or 
*—  -,  and  you  will  have  the  furn  required. 

Note.  Iftbejum  confifls  of  two  terms,  it  is  called 
a  binomial^  or  refidualjurd>  as  tbejign  is  -f-  or  —, 

EXAMPLES. 


+jv/9X3  =  a  3v/2  +  3  v/j: 
j^-jv^Xjlr  ,  2  y    9X3\ 
"^  3     36X1! 


added  to  —\Sxy—y*—i/ax  —  \/xy—y*. 
SECT.      III. 

Of  SUBTRACTION   of  SURD    QUAN 
TITIES. 

CASE 


CASE    L 


all. 


Wloen  the  radical  fign  and,  quantity  are  the  fame  in 


RULE. 


FIND  the  difference  of  the  rational  parts,  to  which 
annex  the  common  irrational  or  furd  quantity,,  with 
the  fign  of  multiplication. 


EXAMPLES. 


80-45=:  y/  16  X  5— 
=  4  — 


-  4X5   -~3  X  5 


3  X  5ft  — 


4x5 


_i2^  —  IOJK 


16x5 


C  A  S  E     IL 

When  the  irrational  farts  are  not  the  fame  in  all. 
RULE. 

CHANGE  the  fign  of  the  quantity  to  be  fubtraftcd, 
the%cpreflioa  connected,  is  the  difference  required. 

EXAMPLES, 


EXAMPLES. 


Vf*  Jubtratied  from  80%  =r  %/  1  6  X  5 


4V/5 

6  ^  16*  ;  34v/g£  —  2  fubtrattedfrom  + 


SECT.    IV. 

Of     MULTIPLICATION     of     SURD 
.  QUANTITIES. 

SURDS  being  confidercd  as  powers  having  fra&ion- 
al  exponents  •,  it  therefore  follows,  that  to  multiply 
one  furd  with  another,  is  to  add  their  fractional  expo 
nents  together,  making  the  denominator  of  their 
fum  the  radical  fign,  and  the  numerator  the  index  of 
the  root. 

HENCZ  is  deduced  the  following  rule  for  multi 
plication  of  furds. 

RULE. 

1.  REDUCE  the  indices  of  the  furds  to  a  common 
denominator. 

2.  ANNEX  the  product  of  the  furds,  to  the  prod- 
udfc  of  the  rational  parts  with  the  fign  of  multiplica 
tion  i  and  it  will  give  the  product  required. 


EXAMPLES. 


(      293      ) 

EXAMPLES. 


=4  6N/32i  z* +.XT Xz' +.T lf = 


SECT.      V. 

O/    DIVISION    of    SURD 


RULE. 


1.  REDUCE  the  furds  to  the  fame  in^ex. 

2.  DIVIDE  the  rational  by  the  rational,  and  to  the 
quotient  annex  the  quotient  of  the  furd  quantities  ^ 
and  it  will  be  the  quotient  required. 

Note.  If  the  quantity  is  thejame  in  both  faftoKS, 
they  are  divided  byJubtraEling  their  exponents. 


EXAMPLES, 

f*  ..  -  -  2, 


Pp 


294 


* 


i  _i_  m  —  n 

fin    ~r   d™  1ZL   ^    mit  • 


SECT.      VI. 

Of    INyOLUflON-  of    SURD   .QUAN 
TITIES. 

THE  powers  of  furds  are  found  by  the  following 
RULE. 

INVOLVE  the  rational  part,  as  in  involution  of  num 
bers  ;  and  to  the  refult  annex  the  power  of  the  furd, 
found  by  multiplying  its  exponent  with  the  expo 
nent  of  the  power  required. 

EXAMPLES. 


of 
'v/36.     The  cube  of  v/3=37^^if  3^  n  \/3 

The  fquare  o 


ir4  X  A;T  zz  4  3  \/  ^+.    -7*^  cube  cf  3  >/^  —  'bx  — 

^~bx\3     3~ax  —  bx  \  Therefore,  when  the  in 
dex  of  tha.power  required,  is  equal  to,  or  a  multiple 
of  the  exponent  of  the  root  -,  the  power  of  the  furd  be 
comes 


(      295      ) 

comes  rational.  The  cube  of  a  —  x\     T—  ^~^]"^ 

=  a  —  x\     T  —  a  —  x]      2.     The  n  power  of  fZ  = 


IF  the  propofed  furd  is  a  binomial  or  refidual  one, 
involve  it  as  in  chapter  vn. 


Thus,  thejquare  of\/6  -f  2v/*'~ 

*  &j  //??<?  operation  annexed.) 

OPERATION. 

\S6+ 
\/6+ 


4^ 

S  "E    I11  "T.1    VII. 
Of    EVOLUTION    of    SURDS. 

THE  powers  of  furds  are  found  by  multiplying 
their  exponents  with  the  index  or  exponent  of  the 
power  to  which  they  are  to  be  involved ;  as  we  have 
Ihewn ;  confequently,  ifthofe  exponents  be  divided 
by  the  index  of  the  root  to  be  extracted,  the  quotient 
will  be  the  exponent  of  the  root  3  which  gives  the 
following 

RULE. 

EXTRACT  the  root  of  the  rational  part,  as  in  com 
mon  extraction  of  roots ;  and  annex  the  root  of  the 
furd,  found  by  dividing  the  index  of  the  furd,  by 
the  index  of  the  root  required, 

EXAMPLES, 


EXAMPLES. 


- 

root  oj  \/a~  a*  '  J-s=  a6  — 


cule  root  of  ^3  =    V 


T  6  v/3- 

Jquare  root  of  3  ^a3 


foot  of  x     r  ~.^      *iz7  x  a  .  If  the  pro- 

V/  ^v 

pofed  furds  are  binomial,  refidua!5  or  trinomial,  &:c. 
find  their  roots  as  in  Chap.  vnr. 

'The  Jquare  root  ofx*-\~6x*  \/y  +  $y  =  ^ 


20  +  i\    x  -~  x  —20  -f  2  vtf  4- 


CHAP.     XL 

Of    INFINITE    SERIES. 

AN  infinite  feries,  is  formed  from  a  fraction 
whofe  denominator  is  a  compound  quantity, 
by  dividing  the  numerator  by  the  denominator ;  or 
the  extra&ing  the  root  of  a  furd  quantity,  which  if 
continued  in  either  cafe,  would  run  on  fempiternal- 
ly  ;  that  is,  the  number  of  terms  in  the  feries  would 
be  infinite  ;  but  by  obtaining  a  few  of  the  firft  terms 
of  the  feries,  you  will  eafily  perceive,  what  law  the 
feries  obferve  in  their  progreflion ;  by  which  means 
you  may  continue  the  feries  by  notation  as  far  as  you 
pleafe,  without  an  aftual  performance  of  the  whole 
operation  at  large.  P  R  O  B, 


PROBLEM    I. 

fo  find  an  infinite  furies  by  dhifion  ;  that  isy  to 
Tfrow  a  compound  fractional  exprejfion  into  fuch  afe- 
ries,  whofe  fum,  if  the  number  of  terms  were  continued 
adinfinilum,  would  be  equal  to  the  given  frattional  ex* 
preffion. 

RULE. 

DIVIDE  the  numerator  by  tkc  denominator  until 
you  have  3,  4,  5,  or  more  terms  in  the  quotient. 

EXAMPLES. 
Throw  — i—  into  an  infinite  feries. 


OPERATION. 


OPERATION. 

I  V  eU'i          V 


V 

o  — - 

y 

V         *V" 


z          V 


v 

•*    _      _ 


HERE  the  law  of  the  progre/lian  which  the  feries 
obferve,  is  plain  ;  for  each  fucceeding  term  is  pro 
duced,  by  multiplying  the  preceding  one  with  — 

~  :  Thus,  the  firft  term  of  the  feries  is  1, which  be- 
y  y 

ing  multiplied  with  —  -,  gives for  the  fe- 

cond 


*99 


cond  term,  and  —  —  X  —  2  =  .!L.=  third  term  i 

y*         y     y\' 

IT         ^*     v  *Z>  1>3        7  ,    -      T  ,     .      ,. 

alio,  —  X  —  ~—  —  the  4th  term,  which  mulnpli- 

-V  3  y  y  4- 

ed  with  ~~  will  give  the  5th  term;  and  fo  on, 
multiplying  the  preceding  term  by  the  common  ratio 
—  ~,  you  may  find  any  number  of  terms  at  pleafure. 

y  ' 

BUT  in  oriler  to  have  a  converging  feries,  or  a 
feries  wherein  the  terms  continually  decreafe,  the 
greateft  term  of  the  divifor  muft  fland  firfl  in  the  or 
der  of  arrangement;  for  fuppofe  in  the  above  exam 
ple,  that  y  is  very  great  in  refpecl:  of  v  ;  £hen  will 

—-.  be  very  great  in  refpecl:  of  —  ;  fo  that  in  this  fup- 

pofition,  the  terms  being  multiplied  with  the  powers 
of'y,  and  divided  by  thofe  of  y\  it  follows,  that 
each  fucceeding  term  is  very  little  in  refpecl:  of  the 
preceding  one,  and  confequcntly  the  feries,  a  con 
verging  feries.  Again,  puti;  for  the  firft  term  of  the 
divifor  (the  fuppofition  the  fame  as  before  )  and  the 

i       v      y3" 
feries  will  be  --  Z,-f^_     &c.   and  fince  y  is  very 

-    v     vz     v3 

great  in  refpecl:  of  v  -,   it  follows,  that  ~  is  very  lit- 

i) 

tie  in  refpecl  of  —,  and  —  very  little  in  refpeft  of 

v*          v2- 

y* 

•^.  ;    confequently,    the  feries  is  a  diverging  one  ; 

that  is,  a  feries  whofe  terms  continually  increafe, 
and  therefore,  the  farther  you  proceed  in  them,  the 
farther  you  will  be  from  the  truth.  Hence,  &c. 


AND  fince  it  is  impoflible  to  afTign  an  infinite 
number  ;  it  follows,  that  the  number  of  terms  ex- 
prefling  the  true  value  of  fuch  a  feries,  is  not  aflign- 
able;  yet  the  taking  of  a  few  of  the  firft  terms  will 
be  fufficient  for  any  practical  purpofe. 

Throw  >a    ••  into  an  infinite  feries. 
v  —  d 

OPERATION. 


v 


,  W_^v 

V  V* 


V3 

HERE,    each  preceding  term,  after  the  firft,  is 

multiplied  with  -,  and  theprodudb  is  the  next  term 

v 

following;   therefore,  the  law  of  the  progreltion  is 
manifeft. 

Throw  -  —  into  an  infinite  feries. 

OPERATION. 


OPERATION. 


—  b* 


HERE  thelaw  of  the,  continuation  is  the  preceding 
terms  multiplied  with"—  fr*. 

PROBLEM      II. 

fo  cxtraft  the  root  of  a  compound  Jurd  in  an  infinite 
Jeri&s  ;  that  /V  ,  to  throw  a  compound  JUT  d  quantity  into 
*  converging  feries,  whofefum,  if  the  terms  were  infi~ 
nitely  continued  would  be  equal  to  the  root  required. 

RULE. 

EXTRACT  the  root  of  the  quantity,  as  in  common 
algebraic  extraction  ;  the  operation  continued  as  far 
as  is  thought  neceflary,  will  give  the  feries  required, 

V 

EXAMPLES. 


Throw  \/a*  +yi  into  an  infinite  feries. 

OPERATION. 


OPERATION. 
«•+,.  (,+£_ 


4*1 


1—  21.4- 

»  *    r 


Tbat  is,  T^+lc^—a  +  --  -f  —  +&e, 


Find 


Find  the  value  of  i  —  #2]r  in  an  infinite  ferics, 
OPERATION. 


--— 

2      8      16 


—  X 


4 


4 

A»*  V6  V   * 

:L  4-  IL.  4*2- 
4        8    r64 


16  8        64 

—  il-4-  ^*  4-  *  '  °  4-  >v  '  * 
"  8""  I61"  64       256 


PROBLEM    III. 

To  reduce  any  fur  d  or  fractional  quantity  into  an  in 
finite  feries,  by  the  celebrated  Binomial  Theorem,  invent 
ed  by  that  Prince  of  Mathematicians,  the  illuftrious 
Sir  ISAAC  NEWTON,  which  is  as  follows. 

Binomial 


(      3Q4      ) 

Binomial  theorem* 


Wherein  it  is  to  be  obferved,  that  P  +  PQJs  the 
quantity  whofe  power  is  to  be  thrown  into  an  infi 
nite  feries  ;  P  reprefents  the  firft  term  of  the  propof- 
ed  quantity  5  Qjthe  other  terms  divided  by  the  firft  -, 

—  the  index  of  the  power,  whether  it  be  affimative  or 
n 

negative  :  And  A  —  firfl  term  of  the  feries  *,  Bthc 
iecond  j  C  the  third  ;  D  the  fourth  ;  E  the  fifth  ; 
F  the  fixth,  &c.  that  is,  the  feyeral  terms  of  the  fe 

ries,    are  A=P7>  B  =2  AQ,  C  =^Z^  BO  .  D 

n  2ft 


EXAMPLES. 

ir 

Reduce   a*+x*\*  into  an  infinite  feries. 


Here  <?*=:?,  ~  ~  Cj,  m~\>andn~i  : 


Mere/ore,  A  =  Pr=  a>  B  n-AQ=:  —  ,    C 


~r  +  ^~,  fSe.  isthejeritsrc 

quired* 

Expand 


(     3Q5      ) 

T  —I-'"   I 

Expand  -77-  =  i+a|        into  an  infinite  fcries. 

T;  therefore,  m~~-i, 


f    m  \ 
i  A~^pT  J 


Find  the  value  of  —  ;  —  in  an  infinite  feries. 
a+y 

Here  ~-  -^X^+jF  *  '>  IWerefore,-'?  =.a, 

y  —  T         * 

—-,™  —  —  i>anJn=:i  :  Wen,  A—  a     *,  or  j 


_  !          i      y     y*     y* 
That  is,  vX^+j]      =  ^  X  -—  ^  +  JT  —  2? 

w         v      vy      vy*      vy* 


PROBLEM    IV. 

Tofind  tbefum  of  an  infinite  feries,  geometrically 
decrcafin?. 

R  U  L  E. 

DIVIDE  the  fquare  of  the  firft  term  by  the  differ 
ence  between  the  firft  and  fecond,  and  the  quotient 
will  be  the  fum  required,  THUS, 


(      306      ) 
THUS,  the  fum  of  the  infinite  ierics  ^—  *a.+  ^ 

V 

fL    &?£.  zr  "y1  -r/y-~^»   and  the  fum  of  v 


/•       -r     - 

v  $  for  u  v 


be  divided  by  v  -f  0,  and  *v  by  ^  —  T;,  the  quotients 
will  be  the  feries  propofed.  Therefore,  the  rule  is 
manifcil. 

EXAMPLES.  , 


Given  i  +  T  +  i  +T>  &c.  ^  infinitum>  to  find 
their  fum. 


,  i  a  -r-  i  —  4-  =  2  the  fum  required. 
Given  T6^-  4-  T!^  4~  WW>  ^c.  adinfinitum^  to  find 
their  fum. 


,    J:    4-  S  — T-k-  =  1  tbejum  required. 

i  ol 

Given  2  —  J  +  *.  —  T*T,  &c.  adinfinitum^  to  find 
their  fum, 

,    4  ^-  2  +  -J  =  1 1  "=.Jum  required* 


CHAR     XII. 

Of     PROPORTION    or     ANALOGT 
ALGEBRAICALLY      CONSIDERED. 

WHEN    quantities  are   compared  together 
with  regard  to  their  differences,  or  quotients, 
their  relations  are  exprefled  by  their  ratios.     The 
relation  of  quantities,  arifing  from  the  firft  compa- 

rifon 


(     3°7     ) 

nfon,  is  exprefTed  by  an  arithmetical  ratio,  that  of 
the  fecond,  by  a  georrietri.caFratio ;  and  the  quanti 
ties  themfclves  are  faid  to  be  in  arithmetical,  or  ge 
ometrical  proportion,  as  the  ratios  of  their  compa 
nion  are  arithmetical,  or  geometrical :  Which  pro 
portions,  together  with  fuch  others  as  arife  from  the 
alternation,  converfion,  &e.  of  thofe-  proportions 
that  are  of  any  conQderableufe  in  Mathematics,  will 
be  noticed  in  the  following  order. 

SEC     T.     I. 

Of  ARITHMETICAL  PROPORTION. 

WHEN*  quantities  increafc  by  addition  or  fubtrac- 
tion  of  the  fame  quantity,  thofe  quantities  are  in 
arithmetical  proportion  :  Thus,  a,  a  -f-  d>  a  4-  id, 
a  +  3<t>  &c,  or  A-,  x — d,  x — 2^,  x — 3  </,  &c. 
are  quantities  in  arithmetical  proportion  ;  wherein 
the  quantity  df,  which  is  continually  added  or  fub- 
tra&ed,  is  the  common  difference  of  the  feries  ;  there 
fore,  when  in  any  four  quantities,  the  difference  be 
tween  the  firft  and  fecond,  is  equal  to  the  difference 
between  the  third  and  fourth,  thofe  quantities  are  in 
arithmetical  proportion  ;  as  in  thefe,  jy,  y — #,  y  — 
2  »,  y  —  3  n  ;  where  y  — y  —  n  ~  n  ;  and  y  --  2  n 
— y  —  3  »  :=  n.  Therefore,  &c. 

THEOREM     L 

If  three  quantities  be  in  arithmetical  proportion,  tie 
Jum  of  the  two  extremes  will  be  double  the  mean. 

Thus  if  a>  a  4-  d,  a-\-  2^  are  in  arithmetical  pro- 
portion,  then  will  a  4-  a  -i -  2  d  n  a  -\-  d  -\^a  4-  d^ 

T  H  E  O. 


(      3Q*      ) 

THEOREM    II. 

If  four  quantities  be  in  arithmetical  proportion,  the 
film  of  the  two  extremes  will  be  equal  to  thejum  of  the 
two  means. 

Thus,  if  /*,  a  +  ^>  *~4~  2  ^  a  +  3  ^  are  quanti 
ties  ia  arithmetical  proportion,  then  will  a +0  -f~3*/;z 

THEOREM    III. 

/«  ajeries  of  arithmetical  proportionals,  thejum  of 
the  two  extreme  terms,  is  equal  to  thejum  of  any  two 
tsrms  equally  dift  ant  from  the  extremes. 

Let  the  feries  be  a,  a  +d,  a+zd,  a+$d,  a+$d9 
&c.  to  z :  Under  which  write  the  fame  feries  with 
their  order  inverted;  then  adding  thofe  terms  toge 
ther  which  {land  directly  oppofite  each  other,  and 
the  fum  of  any  two  fuch  terms,  will  be  equal  to  the 
fum  of  the  firft  and  laft  terms,  as  plainly  appears  by 
the  following 

EXAMPLE. 


toz 
Series  inverted,  z,z  —  £/,a  —  2  d,z  —  3  d3z  —  ^&£$€.  to  a 


thejum  of  every  two  terms. 

Now  from  this  example,  a  rule  for  finding  the 
fum  of  all  the  terms  of  any  arithmetical  feries,  may 
be  eafily  deduced  ;  for  it  is  plain,  that  the  fum  a-j-  z 
~}-a-}-z~^a-±Z)  &c.-or  *2-fz,  taken  as  many  times  as 
there  are  number  of  terms,  is  double  the  fum  of  the 
feries  a,  a  >\-d%  a-j-id,  &c.  Ccnfequently,  that  fum 

divided 


(      309      ) 

divided  by  a,  will  be  equal  to  the  fum  of  the  fcries  ; 
that  is,  (puting  »  =  number  of  terms,  and  j  = 


r  .    r  ,   \a  -f  z  X  n     na  +  nz  ^   . 

of  the  lenes)  —  >  —  ,  -  ~  -  1—  =  s  :  Or  in  words, 

2  2 

the  fum  of  the  firft  and  laft  terms  multiplied  with 
half  the  number  of  terms,  will  give  the  fum  of  the 
feries. 

BUT  in  any  arithmetical  feries,  the  co-efficient  of 
the  common  difference  (^)  in  any  term,  is  i  lefs 
than  the  number  of  terms  to  that  place  ;  confequent- 
Jy,  its  co-efficient  in  the  laft  term,  is  equal  to  the 
number  of  terms  lefs  i  ;  and  therefore,  the  laft  term 
z~a-{-n  —  i  X  ^  —  #  -i-  dn  —  d.  Confequently, 


a  theorem   for  finding. 

2 

the  fum  of  any  arithmetical  feries,  when  the  firft  term, 
common  difference,  and  number  of  terms  are  given, 
And  univerfally,  puting 

a  :n  firft  term  of  an  arithmetical  feries, 

d  zz  common  difference) 

I  n:  laft  term, 

n  zi  number  of  terms, 

s  ~Jum  of  all  tbejeries. 

THEN  having  given  any  three  of  thofe  five  quan 
tities,  the  reft  may  be  found  by  the  following  theo 
rems. 


Theorem  i .  '±4-?  =  s.    Theorem  2.  ;_  =  » 
2  /-M 

Theorem  7.  -Hlf  =  </.    Theorem  4.  ?s      na  = 
w —  i  » 

R  r  Theorem 


(      3*0     ) 

^  r  -  j 

Theorems,  *s~~nl=a.    Or,  »-^Zf  +  I.  /- 
7*  ^ 

nd—d+a.  a~l-\-d — nd. 

SECT.      II. 
Of    GEOMETRICAL    PROPORTION. 

WHEN  of  four  quantities,  the  product  of  the  two 
extremes  is  equal  to  the  product  of  the  two  means ; 
thofe  quantities  are  in  geometrical  proportion :  As, 
a,  ary  b,  br  -t  where  ay,br~ary>b  :  Alfo,  when 
quantities  increafe  with  a  common  multiplier,  orde- 
creafe  by  a  common  divifor,  as,  #,  ary  arz>  ar3 ,  ar4, 

&c.  and  a>   ~,   — ,   — ,    — ,    &c.  thofe  quantities 
r     r*     r3      r4 

arc  faid  to  be  in  gepmetrical  proportion  continued, 
•where  the  common  multiplier  or  divifor  r  is  the  com 
mon  ratio. 

T  H  E  O  R  E  M    I. 

In  any  Jenes  of  quin  titles  in  geometrical  proportion 
continued^  the  firft  term  hath  the  fame  ratio  to  the  Je- 
cond)  as  the  fecond  hath  to  the  third,  and  as  the  third 
to  the  fourth,  &c. 

TiiusJn^r^rV^V^&c.  and  #,  ~,   ~,   -^. 

r     r*     r*' 

-a 

~>   &c.  a  :  ar\\  ar  \  ar*  ::  ar%  \  ar*  :;  arl  \  ar*  :: 

a       a      a         a       a         a       a 

cue.    and    a  :-::  -  :  -r  ::~T:~T::  —  :—  :: 
r      r     r%       r*     r3       r3     r+ 

&c.  For,  ax*r*  rr  arXar,  and  aXar4  —  arXar5  5 
allb,  aX~  ~  f  xf?  and  fo  Qn  for  the  reft> 

T  H  E  O. 


C      3"      ) 


THEOREM    II. 

In  a  feries  of  geometrical  proportionals  continued,  the 
produft  of  the  two  extremes,  is  equal  to  the  product  cf 
any  two  terms  equally  dift  ant  from  the  extremes. 

THUS,  in  the  feries  a,  ar,  ar*,  tir3 ,  ar*,  &c.  If  x 
be  the  laft  term,  then  will  7  be  the  laft  term  but 

one,  and  —  the  laft  term  but  two  5  wherefore, 

~  ax,   the  product  of  the  two  extremes,  and 

sc       o.rx 

-  =  -^-  =:  ax  the  product  of  the  fecond  and  laft  term 

M 

but  one  :  That  is,  ^v^rr  ary^~>  in  like  manner, 

r 

X 

tfX#  =  fff"*X"T>  and  fo  on  for  the  reft. 

THEOREM    III. 

tfhefum  of  any  feries  of  quantities  in  geometrical  pro 
portion  continued,  is  obtained  by  multiplying  the  laft 
term  by  the  ratio, and  dividing  the  difference  between  that 
pro  duff  and  thefirft  term,  by  the  ratio  lefs  i. 

THUS,  let  the  feries  whofe  fum  is  required,  be 
a  +  ar  4-  ar*  +  ar3  4-  ar*,  which  multiplied  with  r3 
gives  rfr4*#r14-flr34-^4+^r5>  from  which  fub~ 
tract  the  former. 

Thus,  I 


Now 


Now  it  is  plain,  that  the  difference  ar3  —  a  is  equal 
to  thefum  of  the  propofed  ferieslriultiplied  by  r  —  i  ; 
confequently,  the  fame  divided  by  r  —  1  3  will  give  the 
fum  of  the  feries  required:  That  is,  (puting  Jzrz 
ar5  —  a 


Or,  generally  ar-{-ar*+ar'i+ar*,&c.  +^+77 

-  ar1,  &c.  +      +~ 


3G          X         V 

'+  "7+  T  +  ~«     That  is,  the  fum  of  any  geometrical 

feries  wanting  the  firft  term,  is  equal  to  the  fum  of 
the  -fame  feries  wanting  the  laft  term,  multiplied  with 
the  ratio.  Wherefore,  s  —  a~s—  .vX*"  3  that  is, 
s  —  a--sr  —  rx,  and  sr  —  s'~:rx  —  a  :  Hence,  s^=z 

r..x  a>.  And  fmce  r  is  not  in  the  firft  term  of  the 
r  —  i 

feries,  it  follows,  that  in  the  laft  term,  its  exponent 
will  be  i  lefs  than  the  number  of  terms  3  and  there 

fore,  (puting  n  ~  number  of  terms)  x  ~  ar 
Confequently,J~{by  writing  for  #  its  equal  ar  "  "  ) 


a  .  And  univerfallv,  puting 


r  —  i  r 

a-=:frft  term  of  a  geometricalferies, 


term, 

s-=Jum  of  the  feries. 

THEN  having  given  any  three  of  the  aforefaid 
quantities,  the  reft  may  be  readily  found  by  the  fol 
lowing  theorems,  which  are  deduced  from  the  above 
equation, 

Theorem 


(      3*3      ) 

rl-a 
Theorem  i.  y_     ~  s. 

2.  rl+s  —  sr~a. 
s  —  a 


sr—s+a 
4-  '    —  -  =  /• 

THEOREM    IV. 

j/7*  /b#r  quantities  are  proportional,  as  a  \b  \\c\d\ 
then  will  any  cf  the  following  forms,  alfo  be  proportion^ 


Direftly,  a  :  b:\c\d. 

Alternately,        a  :  c  :  :  b  \  d. 
Inverfely,  l\a\\d\  c.  _ 

Compoundedly,  a-\-b\  b\\  c+d  \  d. 
Dividedly,  a  \  b  —  a\\c\  d  —  c. 

Mixtly,  b+a  \  b  —  a  :  :  d+c  \  d—c. 

SECT.     III. 

Of    HARMONICAL    PROPORTION. 

HARMONICAL  proportion  arifes  from  the  compari- 
fon  of  mufical  intervals,  or  the  relation  of  thofe  num 
bers  which  affign  the  lengths  of  firings  founding 
mufical  notes. 

THE  moil  ufeful  part  of  this  proportion  in  practi 
cal  Mathematics,  is  contained  in  the  following  theo 
rems. 

T  H  E  O. 


THEOREM    L 

If  three  quantities  be  in  harmonica]  proportion,  the 
firft  will  be  to  the  thirdy  as  the  difference  between  the 
firjl  andfecondy  to  the  difference  between  thejeccnd  and 
third. 

THUS,  if  ay  b  and  r,  be  in  harmonica!  proportion, 
then,  as  a  '  c : :  b  —  a  :  c  —  b  :  Confequently,  ac — ab 
~cb —  ca,  by  multiplying  means  and  extremes: 
From  which  equation  is  deduced  the  following 
theorems. 

cb  <zac 

Theorem  i. 7—0.  Theorem  2. — ; — ~£, 

ic — b  a-\-c 

Theorem  3.  ~— jfe^ 

THEOREM    IL 

If  four  quantities  be  in  harmonica!  proportion,  the 
fir  ft  will  be  to  the  fourth ,  as  the  difference  between  the 
frft  and  JecQnd>  is  to  the  difference,  between  the  third 
end  fourth. 

THUS,  if  the  quantities  a>  b,  c>  a7,  are  harmonical 
proportionals,    it  will    be,    a  :  d : :  b  —  a  :  d  —  -  c  : 
Wherefore,  ad —  ac  zr  db  —  da.   From  which  equa 
tion,  we  get  the  following  theorems. 
db 


(     -3*5      ) 


CHAP.     XIII. 

Of    SIMPLE    EQUATIONS. 

AN  equation  is  an  expreffion,  averting  the  equali-" 
ty  of  two  quantities,  which  are  compared  to 
gether  by  writing  the  quantities  with  the  fign  of 
equality  between  them.  Thus,  if  #-4-3  is  <equal  to 
2#— i,  the  equation  is  formed  thus,  x +3  =  2ff— n 
Alfo,  8 — 3=15  —  10, 

A  SIMPLE  equation,  is  an  equation  which  in 
volves  one  unknown  quantity,  without  including 
its  powers.  Thus,  3* — 1  =  2x^2  is  afimpleequa- 
tion  which  exprefles  the  value  of  the  unknown  quan 
tity;  when  that  quantity  Hands  alone  on  one  fide  of 
the  equation,  the  reft  being  on  the  other  fide,  which 
if  known,  we  then  have  a  determined  value  of  the 
unknown  quantity  in  known  terms.  And  the  bull- 
nefs  of  bringing  the  unknown  quantity  to  (land 
alone  on  one  fide  of  a  fimple  equation,  is  called  re- 
duclion  of  fimple  equations  :  To  effect  which  pur- 
pofe,  we  have  the  following  rules. 

RULE    I. 

ANY  quantity  may  be  taken  from  one  fide  of  an 
equation  and  placed  on  the  other,  if  you  change  its 
fign.  Or  which  is  the  fame  thing,  iubtraft  the  quan 
tity  from  both  fides. 

FOR, 


(      3*6      ) 

FOR,  if  from  thofe  quantities  which  are  equal, 
there  be  taken  the  fame  quantity,  what  remains  will 
be  equal. 

EXAMPLES. 

Given  x — 6  —  20,  to  find  the  value  of  AT. 
Thus,  x~  20-}- 6,  per  rule,  and x—  26  by  addition. 

For,  —  6  taken  from  x  —  6,  leaves  xy  and  —  6  taken 
from  20,  leaves  20  -j-  6,  or  26,  by  the  nature  offub- 
traffion.  Therefore,  &c. 

Given  #-1-4—30 —  5,  to  find  the  value  of  A?. 

fbuSy  x  zz  30  —  5  —  4  by  tranfpofition  : 

Or,  #*~  30 —  9  IT  21  by  addition  andjultraftion. 

If  x — 3.4-1  —  21  : 

fben  will  x  —  2 1  -f  3  —  \  by  tranfpofttion  : 
Or,  ^"d  23  ^jy  addition  andjubtraftion. 

RULE     IL 

WHEN  the  unknown  quantity  is  multiplied  with 
any  number,  it  may  be  taken  away  by  dividing  all  the 
reft  of  the  terms  in  the, equation  by  it. 

FOR  if  thofe  quantities  which  are  equal,  be  divid 
ed  by  the  fame  quantity,  their  quotients  will  be  e- 
qual. 

EXAMPLES. 

Given  47  —  1 2  z=  2 y  -j-  45  to  find  the  value  of y. 

Firfty  ^y  —  27=  12  +  4  by  tranfpofition  : 
Then,  2y.~  16  by  addition  andjubtraffiion  : 

*6 

Or,  y  rrz:  —  —  8,  per  rule. 

If 


(     3*7      ) 

If  6y  +  3=^4.  18,  then  will  6y  —  y  z:  18 
tranjpqfition  ;    <ta*/  5JXZ=  15  byfubtraftibn. 


Whence,  y  ±=  —  =:  5  £y  dwifion. 

>j 

Let  3*  —  10  zr  20  —  #4-6,  be  given  to  find  #, 
^Vr/?,  3*  -|-  ^  z:  20+6  +  10  by  tranffqfition  : 

36 
Or,  4,r  =36,  ^7»rf/  therefore,  x  =  1—  n  9. 

RULE    III. 

WHEN  any  part  of  the  equation  is  divided  by  any 
quantity,  that  quantity  may  be  taken  away  by  mul 
tiplying  all  the  reft  of  the  terms  by  it  ;  which  is  the 
fame  as  to  multiply  all  the  terms  in  the  equation 
by  that  quantity.  And  if  thofe  quantities  which  are 
equal,  be  multiplied  with  the  fame  quantity,  their 
produces  will  be  equal. 

EXAMPLES. 


1} 

Given,   r  +2  —  10,  to  find  the  value  of  v. 

ttus,  v+ii  —  Soyper  rule  : 
And  v  n  60  —  12  1148  by  tranjfofitionandfultrac^ 
tion. 

y      iy      ? 
Let  ^  +  —  +~  ~  16,  be  given  to 


Men,    12L±-2±  -16  by  addition: 

And  $iy  +  2411512  by  multiffyation  ,• 
S  a 


Whence,  y^-  15  A 


Alfo,    if       +  6= 


4jy—  3.?  =  12  —  8  by  tranffofition. 
W  'hence,  7  =  4. 

RULE    IV. 

IF  any  quantity  be  found  on  both  fides  of  the  equa 
tion,  having  the  fame  fign,  it  may  be  expunged  from 
both.  Alfo,  if  all  the  terms  of  an  equation  be  mul 
tiplied  with  the  fame  quantity,  it  may  be  ftruck  out 
of  them  all. 

EXAMPLES. 

5    then    will    2tf= 


per  rule  : 

And  ix—  A?~2  j  or,  #=2: 

£• 
Alfo,  if  6x+c  ~l+c>  then  will  6x~b,  and  -v~* 


^xa       ixa      xa      da 
Moreover,  if  -  —  +  —  -  —  —  —  —  >    then  will 

C  €  C  € 


And  4^  ~d  by  addition  andjubtraftion  : 

d 

Whence,  #  =  — 
4 

RULE    V. 

IF  that  part  of  the  equation  which  involves  the 
unknown  quantity  be  a  radical  expreflion,.it  may  be 

made 


made  free  from  furds  by  tranfpofing  the  reft  of  the 
terms  by  the  preceding  rules,  fo  that  the  furd  may 
fiand  alone  on  one  fide  of  the  equation  :  Then  take 
away  the  radical  fign,  and  involve  the  other  fide  of 
the  equation  to  the  pow0rwhofe  index  is  equal  to 
the  denominator  of  the  radical  fign. 

EXAMPLES. 


—  20  : 

Then  will  \/x  4-  3  =  20  —  4=16  by  tranfpofi* 
tlon  : 

Andx+s  z:  16  x  16  ~  256  by  involution  : 

Or,  *=  256--  3  =253, 

And,  if  4+  \/Q.X  +  6  —  9  5  then  will  \/ix  +  6 
=  9—4  =  5  by  tranfpofition  : 

And  2X+6  s:  25  by  involution  : 

Whence,  x  ~  —  z:  9^-. 

In  like  manner,  if  3\/^+  3  =  10;  then  will 
?\/^—  io—  3—7  s  andax^z^b  involution  -t  or, 


RULE    VI. 

IF  both  fides  of  an  equation  be  a  complete  power, 
or  can  be  made  fo  by  the  preceding  rules,  it  may  be 
reduced  to  more  fimple  terms,  by  extracting  the  root 
of  both  fides. 

EXAMPLES. 

Given,jy*  +  §y  +  9  _  57  =  87,  to  find  the  val 
ue  of>. 

Fitf, 


Krft>  r  +  6y  +  9  —  87  +  57 

%'hen,  y  -f  3  zz  1 2  £y  extracting  the  root : 
Or,  y  —  12,  —  3  :z:  9  £y  tranj-pofition. 

Given,  p^z  4-  24^  4-  16  —  4^a  +32.7+64,  to 
find  the  value  ofj. 

^'^>  3^+  4  —  iy  -f*  ^  ^  extraffing  the  root ; 
And   3jy  —  27  1:8— 4^x  tranffofition ; 
That  is,  y  ~  4. 

RULE    VII. 

-s. 

ANY  analogy  may  be  converted  into  an  equation, 
by  aflerting  the  produftofthe  two  extremes  equal  to 
the  product  of  the  two  means, 

EXAMPLES, 

If  6  +  AT  :  10  ::  4!  6  5    then  will  36  +  6x  zz  40, 
^j?  multiplying  means  and  extremes,  and  6^=4;    0r, 
4 


And,  if—  :  a  ::  10  :  2;  then  will  —  ~ioa>  and 

O  y 

30  a 
4%  zz  30  a  y  or  Arm — —  • 

4 

And  in  like  manner,  if  6 :  A; —  21:4:5$  then  will 
30  — 4#— 8: 

And  4 A; =30  4- 8  z:  38  s  or.  A;  n— —9^. 

COROLLART.    / 

HENCE  it  follows,  that  an  equation  may  be  turned 
Into  3n  analogy,  by  dividing  either  fide  of  it  int© 

two 


(      3"      ) 

two  fuch  parts,  which  if  multiplied  together,  would 
produce  the  fame  fide  again  ;  making  thofe  parts, 
cither  the  two  means  or  extremes  ;  then  dividing  the 
other  lide  in  like  manner  for  the  other  two  terms. 


CHAP.     XIV. 

CONCERNING    the  extermination  of  un 
known  quantities,  and  reducing  thofe  equations 
Which  contain  them,  to  a  lingle  one. 

PROBLEM      I. 

To  exterminate  two  unknown  quantities,  or  reduce 
two  equations  containing  them,  to  a  Jingle  one* 

RULE    I. 

FIND  the  value  of  one  of  the  unknown  quantities 
in  each  of  the  given  equations,  by  the  rules  of  the 
preceding  chapter.  And  puting  thefe  two  values 
equal  to  each  other,  you  will  have  an  equation  in 
volving  only  one  unknown  quantity  •>  which  equa 
tion  if  a  fimple  one,  is  to  be  refolved  as  in  the  lad 
chapter, 

EXAMPLES. 


,  |  £±-£±*0  }  to 


— .y 

From  the  firft  equation,  we  have  #—-—• 


(      3**      ) 


And  from  tlejecond,  ,vzz 


~-?  -  3Q+3.?  . 


^     /• 
Therefore, 

And  84  —  6yz:6o4-6y  £y  multiplication  : 
Whence  y  84  —  60—  1  2^  : 
Or,  i2yz:24  i 

24  14  —  •  y 

And  there  fore  >  jn  —  z:  2,  and  x— 

14  —  2 
writing  if  or  y  its  equal)—-—  •   z:6. 


Given,  ]  3^-rJ=  ^2    (  to  ^ncj  v  a 


22  —  J 

From  the  firft  equation  >  v~  -  3   andtheanalo- 

\j 
gy   turned  into   an  equation,  gives  $v^zy,  or  v  = 

2y  22  —  y       iy 

-  —  ,  hnd  therefore)  -  —  —  • 
j  \j  j 

Whence  we  get  y  no  —  57  ~  6  y  ly  multiplication  : 

And  \\y  ~  no  : 

no 
Or,  j=—  =  10: 

2^  '2O 

^  zz  —  z:  (^y  writing  10  /ir^  ^'/j  equaT)  -7- 

RULE     II. 

FIND  the  value  of  one  of  the  unknown  quantities 
in  either  of  the  given  equations  ;  and  inftead  of  the 
unknown  quantity  in  the  other  equation,  fubftitute 
its  value  thus  found,  and  there  will  arife  a  new  e- 
quation  having  only  one  unknown  quantity,  whofe 
value  is  to  be  found  as  before. 

EXAMPLES. 


(     3*3      ) 

EXAMPLES. 


Give*,  \  Z  +  y  ~  T°  }  to  find  z  and  y. 

L  z  —  y  =  /  3 


the  fir  ft  equation,  we  have  z~  10  —  y,  m£ 
Juljlituted  for  z  in  thejecond  equation, 

Gives  10  —  ^  ~j  n  7,  0r  10  —  2jx  n:  7  : 

./f»</   2JX  =  10  -  7=3*. 

Or,  7  =  1.5: 

Whence,  z  —  (by  writing  i.$for  y  its  equal}  10 
1.5  =  8.5: 


JO  -J-  2V 

thefirft  equation  z  z:  -  -  -  >  an.d  this  value 

10  +  2y 

Jnlftituted  In  thefecond  equation,  gives  %y  +  -  ~- 

=  65: 

Or,    6y  +  16  +  iy  zr  130  ;  whence,  8^  =  120  : 

120 
Or^  =  ~"-  J5s  andz—  —  - 

5  +  15=20. 

RULE    III. 


IF  the  unknown  quantity  is  of  lower  dimenfion  in 
one  of  the  given  equations  than  in  the  ofher  ;  find 
the  value  of  the  unknown  quantity  in  the  equation 
where  it  is  of  lead  dimenfion,  and  raife  this  value  to 
the  fame  height  as  the  unknown  quantity  in  the 
other  equation  ;  or  on  the  contrary,  Then  compare 
this  value  with  the  value  of  the  unknown  quantity 

found 


C      3*4      ) 

found  from  the  other  equation  ;  and  you  'tfill  have 
a  new  equation,  with  which  proceed  as  before, 

EXAMPLES. 
G^en,  {  J+f,f  .1  &,  }  to  find  v  and  y. 


From  the  fir  ft  equation  vn  10  —  y  ; 
And  therefore,  40*z:  10—  y\*  -—  100  — 
lTto>  100  —  ioy+y't~6o+y'i'  by  rule  \ft. 
Whence,  y~z  by  reduction  : 
Or,  100  —  2qy-f  .>*—  y*  ~  60  by  rule  id. 
V/hence,  4On 


And*u  zrio-*-^t:io 
Given,       *  -       to 


The  analogy  turned  into  an  equation,  gives 
,   which   divided  by   z,  gives  32  zs  4 

=—  • 

"  3  ' 

i6y* 

Whence,  x*    ^r  —  ^--* 
9 

-^  therefore,       ^*  +^*  -.  25  : 
Or, 


P  R  O  B. 


(      3*5      ) 

PROBLEM    II. 

fo  exterminate  any  three  unknown  quantities,  x,  y> 
and  z,  or  to  reduce  threefimple  equations  that  involve 
them*  to  a  Jingle  one, 

RULE. 

FIND  the  value  of  x  in  the  three  given  equa 
tions;  then  compare  the  firft  value  of  x  with  the  fe- 
cond,  and  there  will  arife  a  new  equation  involving 
onlyjx  and  z.  Again  compare  the  firft,  or  fecond 
value  of  x  with  the  third,  and  there  will  arife  another 
equation  involving  onlyjx  and  z  ;  then  proceed  with, 
theie  two  equations  as  directed  in  the  lad  problem 

EXAMPLE.    \ 

r  2.v  4-^4-2?—  1  5    1 
Given,  <  x  4-67  —  2  •=.  29    >  to  find  x,y  and  z, 

(.  4-V—  1Z  -i-  2^—  I  2     J 

T     £      m,rwm-Tn-     O    fX    MMM*   *U 

the  firft  equation,  we  have,  x  ~  -  —  o 

the  fecond,  ^^29  4-2  —  -6.7  : 
1  2  4-  2%  —  2y 


4 

I  C      '•"'  2^2  T--     V 

Whence,  -  -^  -  ^294-% 

i  -2  4-  22  —  ajy 
294-2  —  ojyn 


From  the  firfl  Qfthefe  equations,  we  get  15  —  22  —  - 
—  584-22  —  ioy;  ^  117^:58—  »  154-4.^: 


T  t 


4?+42 
Whence,  y~  \*   : 

From  the  fecond,  we  have  116+42  —  24jyr: 
22 — ay  : 

104  + 
That  is,   22j— 116 —  12+22;  or,yzz - 

104+22      43-|~4z 
Consequently,  -  — —  *—— 

Whence,  1144+222—946  +  882  : 
And^z  —  222—198  : 
That  is,  662—198  : 

^..      _'«98 


— , 
Whence,  y—  — — —  —  — — —  =  5,    and  x  — 


AND  nearly  in  the  fame  manner,  may  be  exter 
minated  any  number  of  unknown  quantifies ;  but 
there  are  often  much  fhorter  methods  for  their  exter- 
mination,  which  are  beft  learned  by  practice  j  yet 
fome  of  them  may  be  thus  generally  given. 

RULE. 

LET  the  given  equations  be  multiplied  or  divid 
ed  by  fuch  numbers,  or  quantities,  that  by  addition^ 
fubtraction,  multiplication,  divifion,  involution  or 
evolution  of  any  two,  or  more  of  the  equations,  one 
or  more  of  the  unknown  quantities  may  vanifh. 
Then  taking  the  refult  and  the  other  equations,  and 
proceed  as  before,  until  you  have  an  equation  in 
volving 


(     3*7      ) 

volving  only  one  unknown  quantity,   whofe  value 
may  be  found  by  the  foregoing  rules. 

EXAMPLES. 


Given,  j  ^gzj?  }  to  find  *  and> 

Multiply  the  firft  equation  with  2,  and  it  will  give 
4,v  +  6y  —  5  8  ,  andthefecond  with  3,  gives  9*  +  6y  n:  93, 
from  which  Jubtr  aft,  4*  +  6y  —  58  ;  and  you  will  have 

35 
5x^:355  cr,  .vzr—  —  7, 

29  —  2y_29—  14_ 
T"  3       ^5< 

r  2^+4^+32=38  i 

Given,  <    3^+57+62  =  63    Sto  find^,jK,  ancl%. 


double  thefirft  equation  Jultraft  the  Jecond, 
and  from  double  thejecond^fubtraft  the  third,  and  the 


refults  will  be,\ 
1 


x  +  3^  = 


dg*in,  from  thefecond  of  theje  equations,  Jubtraft 
thefirft,  and  the  rejult  will  be  x—4  ;  and  from  double 
the  firft  fubtraft  thejecond,  and  it  will  give  3y=9  ;  or, 


y  =  -  ~  3.    And  from  the  firft  of  the  given  equations, 

38  —  2*—  4? 
2  =  38  —  2A?  —  4x5  or>z=  -  -  - 


38  —  8  —  12 
-  —  =6. 


Mifcellaneous 


Mifcellaneous  Examples. 

Given, 
3 


y  r:  3.2 

tfhejirft  equation  involved  to  a  fquare,  gives  v*  + 
*~  144;  tf«^  f*  —  ivy  +y'L~  16  by  fub- 
tracing  \vy  (  n  *  28  )  //^w  /£*  /<^  equation  5  0r;  ^  —  y 

~  ^  by  evolution  :  ___  __      _ 

^f/  therefore,    v  -\-y-\-v  —  j  ~  1  2  -f  4  : 

16 

Or,  2v  ~  1  6  ;  ^;;^'y  —  —  ~8  : 

Again  3  v-\-y  —  v—  j  =  12  —  4: 

8 
=:  8  i  or, 


^  Lto  find  ^  andj. 


i^y  =  H4    I 
^  L 

J  =  9        J 


.F/V/?,  *y  =  9^  by  multiplication  : 
ConJ'equentlyy  vy  ~  $y  Xj^  =  144  : 


14.4 

is,  97Z  z=  144  ;  cr,  jyx  =:  —  =  16. 

Whence,  y~  \/i6  =z  4  ;  ^7^  i;  —  9^  =  36 


-y  =  $6     j 
Given,  "{  ^  _  o  (  to  find  v  and  y. 


{v—y 
-  — 
y  " 


irfti  v  z:  56  +  y  ;  and  therefore,  —     —  ~   8 


or,  56  +  j  =  8j  ;  whence*  y  s:    r  =  85  ^^^/  v  =  8jr 

=  64. 

Given, 


(      3*9     ) 

to  find  v. 


Given,  I  tf  +  /l6"+Tr=  "^T^  £ 


That  is,    vi/ib+v*  4"  164-1^—32: 
ir  16  —  ^a  : 


involution,  v%  X  i  6-f--y*  ~i6  —  v*]*  :n  256 


That  is  ,  i6-i;z-4-'y'tiz:256  —  32<z;a 

Or,  i6v*  —  '2$6  —  J2i;i  ;  and  iSz;2-  4-32^-^ 


«,  v*=          ;  or,  v= 
16     i  i 


C  Va  -4-  V  a  "~"~  ^2  7 

Given,  \  x     --   J~     f  to 


Again,  x  l  —  2^4"^*  —  a  —  *    : 
7*^^,  #—  7  i=.\/  a—ib  :  _ 

'Therefore^  x+y+x  —  jy  ^r  */a  +  2^  +  V^^  — 
,  2AT  n  \/^  4-  2^>  -f  \/  tf  —  2^  : 


Or,  X-  2 

And  x  -f-y  —  x  — y  —  2jr  n:  -v/^/x  4-  2^ — -v/^"" 


.... 

Whence,  y  = 


CHAP, 


(      33°      } 


CHAP.     XV. 

Of  the  SOLUTION  of  a  variety  of  QUES 
TIONS,  that  produce  SIMPLE  EQUA 
TIONS. 

AFTER  forming  a  clear  and  diftinct  idea  of  the 
queftion  propofed  5  the  unknown  quantities 
mult  be  exprefled  by  letters,  which  muft  be  ordered  in 
fuch  a  manner,  as  to  exprefs  the  conditions  given  in 
the  queftion  concerning  thofe  quantities.  Thus,  if 
the  fum  (  s  )  of  two  quantities  (x  and  jy)  are  requir 
ed  -,  then  is  x  -j-jy  —  s,  an  exprefiion  anfwering  that 
condition.  Alfo,  if  the  difference  (d)  of  thofe 
quantities  is  required  ;  that  condition  muft  be  ex- 
prefled  thus,  x  — jy~  d  (x  being  the  greater)  Their 
product  (f)  is  expreffed  thus,  xy  ~p.  Their  quo- 

v> 

tient  (q)  is  -  —  q.     Alfo,  the  fum  of  their  fquares 

(a)  is  expreffed  thus,  #4  -\- y*  —  ay  and   the  differ 
ence  of  their  fquares  (b)  thus,  xz  — jy*  ~bt 

HAVING  expreffed  the  unknown  quantities  in 
equations  anfwering  their  relations,  or  properties,  as 
given  in  the  queftion ";  you  are  next  to  confider 
whether  your  queftion  is  limited  or  not  -,  that  is,  whe 
ther  the  quantities  fought,  are  each  of  them  capable 
of  more  known  values  than  one  •,  which  may  always 
be  difcovered  in  the  following  manner.  If  the 
equations  that  arife  from  expreiling  the  conditions  of 
the  queftion,  are  in  number  equal  to  the  quantities 
fought,  then  is  the  queftion  truly  limited  :  That  is, 
each  df  the  quantities  fought,  cannot  have  more  val 
ues  than  one  in  giving  the  anfwer  :  But,  if  the  equa 
tions 


ticms  exprefllng  the  conditions  of  the  queftion,  are 
fewer  in  number  than  the  quantities  fought,  then  the 
queftion  is  an  unlimited  one  ;  that  is,  the  quantities 
fought,  are  each  of  them  of  an  indeterminate  value, 
and  confequently,  the  queftion  propofed,  capable  of 
innumerable  anfwers. 

AFTER  you  have  discovered  that  the  propofed 
queftion  is  limited ;  you  mud  then  proceed  to  exter 
minate  the  unknown  quantities  by  the  rules  already 
given,  or  other  methods,  which  you  may  learn  by 
practice  ;  to  which  we  now  proceed. 

1.  What  number  is  that,  from  which  if  you  take 
40,  the  remainder  will  be  115  ? 

Call  the  number  /ought  v : 

Then  will  v  —  40  n.  115  by  the  queftion  : 

Or,  v n  1 15  +  40  zz  155  the  number Jought. 

2.  What  number  is  that,  from  which  if  you  take 
10,  and  multiply  the  remainder  with  4,  the  product 
will  be  30  ? 

Call  the  number  fought  v : 

Then  will  v  —  10  be  the  remainder  : 

And  v  —  10X4  —  3°  by  ths  queftion  : 

That  is>  4&  —  40  =  30  : 

Or,  4^  =  30  +  40  —  70  ;  or,  v  n  7T°  —  i?4- 

3.  To  find  two  numbers  whofe  fum  is  80,  and 
their  difference  16.  , 

Let  v  =  the  lea  ft  of  the  required  numbers : 

Then  will  v  +  16  iz  the  greater  by  the  nature  offub- 
trattion  : 

And-v  -f  v  +  16  ~  80  by  the  queftion  : 

That  isy  iv  ~  80  —  16  =  64  : 

Or,  V—  V  =132  ;  and  v+  16^:32  -f-  16^48, 
/bf  greater  number  required, 

4- 


4.  What  number  is  that,  which  if  multiplied  with 
one  third  of  itfelf>  will  produce  the  number  fought  ? 

If  you  call  the  number  Jo  ugh  t  v  : 

i) 
Then  will   —  be  one  third  part  of  '  <v  : 

v 

And  v  Y>-~v  by  the  queftion  : 

v* 
That  is,  —  •  =  v  -,  or,  v*  —  3^,  andv  =z  3  the  num- 

M 

far  fought. 

5.  Suppofe  the  diftance   between    Boflon    and 
York,  to  be  150  miles  ;  and  that  a  traveller  fets  out 
from  Boflon,   and  travels   at  the  rate  of  5  miles  an 
hour  ;  another  fets  out  at  the  fame  time  from  York, 
and  travels  at  the  rate  of  8  miles  an  hour  :  It  is  re 
quired  to  know  how  far  each  will  travel  before  they 
meet. 

Jfyoufut  v  for  the  diflance  that  muft  be  travelled 
by  the  en  e  which  Jets  out  from  B  oft  on,  and  y  the  diftance 
travelled  -by  the  other  before  they  meet  : 

Then  will  v  -f  y  rr  1  50,  the  diftance,  travelled  by 
both,  and  v  I  y  \  :  5  '.  8  by  the  queftion  : 

$y 

That  is>  81;  z=  5^  5  or>  i>zi—  • 

$y 

*AlfO)  v  zz  1  50  —  y  5  conjequently,  —  ^:  1  50  —  y  : 

That  is,  $y  zr  1  200  —  %y  : 
Whence,  y  =  1  200—13  rz  92-^.. 
And  v  =  150  —j>  =  57X9T- 

6.  What  fraction  is  that,  if  you  add  r  to  the  nu 
merator,    the  value  will  be  4-  ;  but  if  you  add  I  C6 
the  denominator  the  value  will  be 


Put  -for  thefrattionjought  : 


(      333      ) 


r,> 


Consequently, 

Or,  3^  —  6  =:  2jy  -f  2  : 

y  ~  8,  /^^  denominator  : 

y  —  2     8  —  2 

rr  3  /^^  numerator  : 


\is  the  fraction  required. 


7.  What  two  numbers  are  thofe  whofe  fumb6o>and 
the  fum  of  their  fquares  2250  ? 

Call  one  of  the  numbers  w>  and  the  other  y  : 

Then -will,  i 


'ion. 


rfty  iv*  +  2  wy  -f-JV*  rr  60^  zz  3600  ; 

And  w*  +  2  wy  -f-  ^r  —  w2"  ~f"^z 
Therefore,  ^wy—  2700  : 

*  -k  2  wy-  -j-^y*  —  -  4  sqy  =r 


Whence,  w  —  ^'rr  v^poo  =r  30  : 
^w^  2  T^  =  60  +  30  =z  90,  or  w  =  45  : 
And  y  =1  60  —  wn:6o  —  45^1  15. 

8.  There  are  three  numbers  in  arithmetical  pro- 
greffion,  the  firft  added  to  the  fecond  will  make  15., 
and  the  fecond  added  to  the  third,  21  :  What  are 
thofe  numbers  ? 

Uu  Let 


(      334      ) 

#,  y  and  z  reprefent  the  three  numbers  : 
Then  will  x  +y  ~  15,   thejum  of  the  fir  ft  andje~ 
d  : 

Andy  +  z  —  2I>  *b*fum  of  the  fetond  and  third  : 
Alfo,  x  -f-  z  zz  2jy  £j>  /£<?  nature  of  the  proportion. 
Whence,  x  4-  jy  -f  ^  +  2;  =  15  -f  21  =36  .- 
That  is,  x  -f-  2j>  -f-  z  ~  36  ;  c?r,  A:  -f  »  —  36  —  2j  : 
5^/  ^  4-  z  s:  2j  s  therefore,   zy  ~  36  —  2^  5  ^r, 
=  36: 

Whence,  y  n:  9  j  ^«t^  ^^115  —  JK  zz  1  5  —  9  —  6  : 
f~  21  —  j^  zz:  21  —-9  n  12. 


9.  Two  merchants  traded  in  partnerfhip  ;  the 
fum  of  their  flocks  was  600  dollars  ;  one's  flock  was 
in  company  8  months,  but  the  other  drew  out  his  at 
the  end  of  6  months,  when  they  fettled  their  accounts, 
and  divided  the  gain  equally  between  them  :  What 
was  each  man's  flock  ? 

Call  one  of  the  flocks  x  ;  then  600  —  x  rr  the  other  : 
But,  x  \  600  -i-  x  :  :  6  :  8  by  the  queflton  : 

8#  nj6oo  —  6  x  ;  or,  14^  n:  3600  : 


3600 
Whence,  x  —  -—  •  =  257^-  ;  and  600  —  x  =  600 

•—  2574-  —  342-f-  /£><?  others  ftock. 

10.  To  find  three  numbers,  fuch  that  if  the  firfl 
be  added  to  the  fecond,  their  fum  will  be  12  ;  and 
the  fecond  added  to  the  third,  their  fum  will  be  20  ; 
alfo,  if  the  firfl  be  added  to  the  third,  their  fum  will 
be  1  6. 

Call  the  firft  number  x,  thefecondy,  and  the  third  z  : 
Then  will  x  -f-  y  zr  12! 

And  y  -f-  z  =  20  V  by  the  queftion. 
Alfo,  x  +  z~  i6J 

therefore,  x  +y  +  x  +  z  ~  1  2  +  16=  28  : 


(335      ) 


fbat  is,  1  x  +^  +  z  r:  28  :   But  y  +  z  ir  20  ; 
Confequently,  ix  4-  20  =  28  ;  0r,  2#  =  8  :'•' 
Whence,  x  =  4,  #»*/  jy  =  12  —  #=12  —  4  ==  8  : 

—  2O  -  T  .==  2O  —  8  :Z  12. 


ii.  There  are  four  numbers  in  arithmetical  .pro* 
greflion  ;  whereof  the  product  of  the  two  extremes  is 
i  r  2,  and  the  product  of  the  two  means  130  -,  alfo, 
the  fum  of  the  firft  aad  fecond  terms  is  vj  :  What 
are  thofe  numbers  ? 

Put  x  for  the  leaf  term,  and  y  the  common  differ 
ence  j  then  will  x,  x  -\-y,  x  +  27,  x  -\-  3^  be  thefou^ 
numbers  required : 

^_  -  ^^_     ., 

by  the  queji 


Whence,  **+  3*y+  27*  —  ^x+3^  =  1  30  —  1  1  2 
=  18: 

,  2jl  ~  18  j  (?r,jya  —  V8  ~  9 


•  +  x  4-j^  r:  i?  ;  ^^  is>'M  +3  =  17: 

Or,  2^rri7—  3  =  14  : 

Confequently,  x  —  V4  =  7^  $*jtrft  term  of  the  $ro~ 
grejfion  ;  ^^  therefore,  x  -f  j  =:  10,  the  JeTond  term  5 
^w^  ^  H-  2jK  =  1  3,  /#<?  third  term  ;  alfo,  x  -f-  3jy  =:'  1  6, 
the  fourth  term. 

So  thatj,  10,  13,  and  16,  are  the  numbers  required. 

12.  There  are  three  numbers  in  arithmetical  pro- 
greflion  ;  the  product  of  the  two  extremes,  is  128, 
and  the  product  of  the  lead  extreme  with  the  mcan^ 
is  36;  \Yhat  are  thofe  numbers  ? 

Call 


Call  the  numbers  required,  v,  y  and  z  j  s>  and  z  be 
ing  the  extremes  y  whereof  v  is  the  leaft. 


y  +  z~  iy  by  the  nature  of  the  proportion  : 

128 
zr  -  from  thefirft  equation  : 

z  z:  iy  «-  v  from  the  third  : 

128. 
Conjequently  y   •s^—  zc  -iy  —  v  by  equality  : 

That  is,  128  7=  'iyv  —  v*.     But  zyv  z:  96X2  rr 
192  : 

tTherefore,  128^:192  —  ^a  by  fubftitution  : 
And  v*  ~  192  —  128—645  or,  v  =z  ^64  ^:  8  : 

128       128 
z  ~  ~-  =  --  zz  16  5    ^^    i;  +  z  —  2y  ; 


8  +  i6 

.  ,,  '• — - —  z:  1 2  j  and  therefore  the  num 

bers  fought  are  8,  12,  1 6. 

13.  To  find  a  fraction,  fuch  that  the  fquare  of  the 
numerator,  added  to  the  denominator,  lhall  make 
30  ;  and  if  2  be  added  to  the  denominator,  the  value 
of  the  fraction  will  be  equal  to  the  reciprocal  of  the 
numerator. 

Put  -for  the  fraftion fought. 

will  v*  +y  z:  30! 

V  i  I  by  the  queftm* 

And      t      —  —  l 

jt  "T"  v  j 

Andy*  —y  4-2X1  —^4"  2  : 
Confequently, y  4-  2  z:  jo—  y  >  -that  is,  27—28*: 

Or, 


(      337      ) 


Or,  y  =  V  —  J4  5  *«<*  ^z  =  3°  —  y~3<>  —  *4 
—  165  or,<v  —  1/16  ±:  4. 

4  4 

So  that  the  fraftion  fought,  is  —  ;  for,  -        »  — 

4=*~-  ;  therefore,  &c. 
16     4     ^ 

14.  To  find  a  number  confiding  of  two  places, 
fuch  that  the  fum  of  its  digits  lhall  be  5,  and  if  9  be 
fubtrafted  from  it,  the  digits  will  be  inverted. 

Let  v  and  y  refrefent  the  two  digits,  v  that  which 
ftands  in  the  tenth's  'place. 

Then  by  the  nature  of  notation,  we  have  lov  +y=z 
ibe  number  fought. 

therefore,  v  +  y  -  5  ?  ^  ^ 
—  9^  xojy  -h  ^3 


And  lO'y-fjx  —  9^  xojy 

n  y  -I-  Q 

Whence,  $v  ~  qy  +9  -,  or,v=.  --  •—  z:  (  ly  di- 

<vijion)  y  +  i  .- 

Alfo,  v—s  —  y  ;  and  therefore,  y  +  i  n  5  —  y  ;  tr* 


Andy—  ±.-=11  •>  ^^^—7+11124-1=3:  St 
that  32  is  the  number  required. 

15.  A  certain  company  at  an  inn;  when  they 
came  to  pay  their  reckoning,  found  that  if  there  had 
been  two  perfons  lefs  in  company,  they  would  have 
paid  a  dollar  a  man  more  ;  but  if  there  had  been 
three  perfons  more  in  company,  they  would  each  of 
them  paid  a  dollar  lefs  :  What  was  their  reckoning, 
and  the  number  of  perfons  to  pay  it  ? 

Put  V  i=  the  number  of  perfons,  and  y  the  number  of 
dollars  each  faid  j  then  will  vy  ~  the  whole  recon~ 
ing. 


(      338      ) 


5*  fy  the  queftion* 


That  isy  vy~vy  +  v  —  27  —  2  from  the  jirft  equa~ 
'tion  : 

Or,  iy  -f-  2  =  v  : 

And  vy~vy  —  v  -\-  $y  —  3  from  the  fecond  equa* 
tion  : 

Or,  v  "=.  3y  —  3  : 

Confequently,    $y  —  3  ~  iy  +  2  ;  or,  3y~-iy  — 

2+  3  : 

Whence,  y  •=.  5,  the  number  ef  dollars  each  p  aid: 
And  v  =  iy  =|-  2  ~  1  2,  w  number  of  -perjjfh  : 

Confequently  ,  vy  n:  60  dollars^  the  whole  reckoning. 
1  6.  To  find  three  numbers  v,  >  and  w,  the  produfb 

of  each  with  the  fum  of  the  other  two  being  given 

viz.   v  X?  +  w  ~="  93°  >  J-X  ^  +  w  =  1300,     and 
w  X  v  +J  =  1480  : 


r  vy  j^  <ww  —  p  jo  rr  ^ 
Or,   \  vy  -|-  ^  n  1300  =  ^ 
t'y^  -f-  «ryzz  1480  =:C 


Then,  vy  -|-  *vw  +  ^^  4-  ^J  zi^z  4-  r, 

rr  ^  .* 

therefore,  zvw  ~a-{-  c  —  b  >,  cr,  vw~ 


^^      ^:  But 


^.a,  +    —  c  -,  or,  vy  ~ 


Again,  vy  -{-wy  -f-  'yze;  +  ^7  ~  ^  +  ^  '• 

v~a  : 

And  therefore,  we  have  i  wy  ir  b  +  c  —  a  \  or,  wy~ 

c 

'2, 


(     339     ) 
.f.  c  —  a 

tmt    "  •    '» 

2 

a  +  b  —  c 
Buty  ~'  ••"      —  i  <?#^  £y  writing  this  value  for  y 


^n  the  equation,  wy  — 


by  redu5lion> 


.  3 

aw  +lw  —  cw  a  -f  c  —  b 

vw  *  -  -  --  ;  or 


a  +c  —  b  aw+bw—cw 

SZ  .....  -  »  and  therefore  by  equality  >  —  .    .  • 

Q.W  b-?  c  —  a 


Wkaue,  w*  =  '\a\^_K 

turned  into  numbers^  and  the  root  extra£fed>  w  will 
befeund  IZ  37  i  whence  the  other  numbers  are  readily 

a-j-c  —  b  a  -4-  b  —  f 

found ;  for  v  z:  —        '    *"  zz  i c,  and  y  z^  •     • 

•<  J  Q.W  JJ          *  2<u 


17.  Two  women  weat  to  market  with  42  eggs, 
for  which  they  received  equal  fums  of  money  ;  af 
terwards  fays  one  to  the  other,  if  I  had  fold  as  many 
eggs  as  you,  I  (hould  have  received  350  cents  ;  fays 
the  other,  if  I  had  fold  no  more  than  you,  I  ihould 
have  received  but  14  cents.  Query,  the  number  of 
«ggs  each  fold,  and  the  particular  prices  fold  at  j  al~ 
fo  the  number  of  csats  each  received* 

Let 


(      340      ) 

Let  17  =s  number  of  eggs  fold  by  one,  and  y  the  num- 

lerfold  by  the  other ;  ajfo,  u  rr  price  which  v  eggs  were 

fold  at  per  egg,  and  w  the  price  that  y  eggs  were  fold 

'per  egg. 


!v+^=4i 
l*w^&o 
yu  =  14 


w 

14 

3S® 

From  the  third  equation  we  have,,  v  n  -  —  :    Prom 

14 
the  fourth  equation,   uzz  —  >    and  therefore,    vu  r^ 

350     14     4900 

-  X  —  m  -  •  But  vu  r:  yw  from  the  fecond  equa- 

4900 
tion  ;  wherefore^  -  irj«;  ;  or  4900  -^y^w7-,  and 

:  /49°o  =  7°  i  ^^^^  ^  =—  :    But  y—^ 

jo  _  14 
the  fourth  equation  ;  confequentlyy  —  —  —  >  or, 


)>  writing  $u  far  w 

in  thefecond  equation,  we  have  vu  zr  $uy3  or  dividing 
loth  fides  by  u,  we  /hall  have  v  ir  5^  :  j5«/  v  ~  42  —  .y 
/r^i»  thefirft  equation  ;  therefore,  $y  31-42  —  ^  ;  cr, 

42  H 


2,    «^  w  =  5//n:  TO. 

1  8.  Given  the  fum  (s)  and  product  (f)  of  two 
quantities,  to  find  the  fum  of  their  fquares>  cubes, 
biquadrates,  &c* 

i^/  v  and  w  reprefent  the  two  quantities  : 
tten  will\   ^*Jf^7"rl   £j  ^  queftion, 

And 


C 


And  x  -4-jyl*  =  #*  +  sry  4-  j*  =r  s*  ly  involution  : 
Or,  'AT*  +  sry  +  jy*  —  3#y  =  s1  —  if  byfubtraff. 
That  is,  x*  +y*~s*  —  ip  —Jim  tifthefquares. 


Again,  x*  +y*  X  x+y  —   *        p  X  s  : 

That  is,  x3  +  xy  X  ^  +^  4-^3  ~  J3 

Or,  x3  +J/>-f-^3riJ3  —  2^/>  ^  writing  sp  for  its 


equal,  xy  x  #  +J  ;  whence,  x 
~j3  —  zsp—fum  of  their  cubes. 


>«/   i,   Jf    +^y  x  ^ 

or,  (by  writing  for  xyy^x*  -\-y*  its  equal,  s^p  — 


zrj4  —  4^a^>  +  2/*  Zl/«»i  of  their  fourth  powers. 

*  X*  +  J—s*  —VP  +  ^  Xs  : 
That  is,  x*  +  *j  X^3  H-jr  +  ys  =*5  —  4*3P 


&p*s  ;  and  therefore,  (by  writing  for  xyXx*  +  jy3  //j 
equal  s^p  —  zsp*)  we  have,  xs  +  s*p  —  jjy*  +  y* 
zzJ5  —  4J3^>  4-  2J[f  a  ;  ^W  ^jy  tranfpofition,  we  get 

4  /?r  tbefum  of  their  fifth 


powers  -,  and  Jo  en  for  the  reft. 


CHAP.     XVI. 

Of   QUADRATIC   EQUATIONS. 

A  QUADRATIC  EQUATION,  is  an  e- 
quation  of  two  dimenfions  involving  only  one 
unknown  quanticy  3  and  is  either  fimple  or  adfeftcd. 

Xx  A 


A  SIMPLE  quadratic,  is  an  equation  which  invol 
ves  only  the  fquar'e  of  the  unknown  quantity.  Thus, 
vz  ~  a*  is  a  fimple  quadratic  equation. 

BUT  when  you  have  an  equation  which  involves 
the  fquare  of  the  unknown  quantity,  together  with 
its  produft  with  fome  known  co-efficient,  you  have 
what  is  called  an  adfefted  quadratic  equation.  Thus, 
v^+av  zi  be,  is  an  adfedted  quadratic  equation. 

ALL  adfefted  quadratic  equations,  fall  under  the 
three  following  forms  : 


f   #*  4-  av  —  be 
viz.   <    v*  -1—  av  zr  be 
(.   vz —  av  ~ —  be 


THE  folution  of  adfedled  quadratic  equations,  or 
finding  the  value  of  the  unknown  quantity  in  thofe 
equations,  is  performed  by  the  following 

R  U*L  E. 

i.  TRANSPOSE  all  the  terms  that  involve  the  un 
known  quantity  to  one  fide  of  the  equation,  and  all 
the  terms  that  are  known  to  the  other  fide. 

£.  I?  the  fquare  of  the  unknown  quantity  is  mul 
tiplied  with  any  co-efficient,  you  muft  carl  off  that 
co-efficient,  by  dividing  all  the  terms  in  the  equation 
by  it,  that  the  co-efficient  of  the  higheft  dimenfion 
of  the  unknown  quantity  may  be  unity. 

3.  ADD  the   fquare  of  half  the  co-efficient  pre 
fixed  to  the  unknown  quantity,  to  both  fides  of  the 
equation  ;  and   that  fide   which  involves  the  un 
known  quantity  will  then  become  a  complete  fquare. 

4.  EXTRACT  the  root  from  both,  fides   of  the  e- 
qtiation,  which  will  confiftof  the  unknown  quantity 
connected  with  half  the  aforefaid  co-efficient ;  and 
therefore  by  tranfpofing  this  half,  the  value  of  the 
unknown  quantity  will  be  determined.          SOL. 


(      343      ) 

SOLUTION    of    the    THREE    FORMS 
Of  QUADRATICS    ILLUSTRATED. 

Let  it  be  required  to  determine  the  value  ofv,  in 
the  fgrm  vz  +  av~bc. 

a*  'a* 

Fir  fly  v*  +  av  +  —  H  be  -\  --  by  adding  thefyr, 

a  a 

of  -  to  both  fides  of  the  equation  :    Then  v  -j-  -  ^ 

a^ 
i/bc  +  —  by  extracting  the  root  of  both  fides  -,  or,  v  == 

a* 

'  --  "ify  tranfpofition.  But  the  fquare  root  of 

any  fofitlve  quantity,  may  be  either  fojitive,  or  nega 
tive  j  that  is  ',  the  fquare  root  of  4-  nz  may  be  either  -f- 
-n  or  —  n  \  for  +  ;/  X  -f  »  ;  or,  —  n  X  —  n}  are  re- 

fpeffively  equal  to  +  n*  .  It  follows  therefore,  that  all 
quadratic  equations  admit  of  twofolutions,  that  is,  the 
unknown  quantity  has  two  values  in  the  given  equation. 
Thus,  in  the  foregoing  example,  where  v*  '+  av  -f- 

a*  a*  a  """«* 

-~  z:  fa  +  —  >    we  may  infer,  that  v  -j-  -  n  \/bc+  —-. 

^',  for,  +  </bc  +  *—  X  +  ^tc  +  - 

4*  T"  T* 


~  X  —  V^c  +  —  are  wf^  equal  to  be 

a* 

-f-  —  »  and  therefore  the  two   values  of  v,  are  v  :=: 

^  a"-     a 

"  —  »   and  v  n:  —  •  i/bc  +  ""    —  :     WHOP 
2  4      ^ 

ambiguity 


(      344.      ) 

ambiguity  is  cxprcjfed  by  writing  the  uncertain  fign  Hh 


before  </bc  +  —  :  ThuSyV-k  ~  iz  ±  v/^  +  — 
4  2  4 


4      2 
exprej/ion  for  the  value  of  v,   viz. 

™     a  .  a_      a* 

+  —  —  ~3    the  only  negative  quantity  is~*~v  — 


which  is  evidently  lejs  than  \/  be  -[-  —  > 

quentty,  the  value  ofv  is  fofitive.  :  But  in  thefecond  ex- 

a7-     a 

frejfion,     viz.     v  =   —  \S  be  4-  ---  ?     having 

~  a*  a 

\/$f  +  —  v    »»$  ~  ^<?/^  negative  -,  if  follows,  that 

the  value  of  v.  muft  alfo  be  negative. 
Again,  if  2*  —  az  —  bc\ 

tfhen  will  z*  —  az  H  ----  ~^-f  —  by  adding  the 

4  4     '       __ 

Jquare  of  -  to  both  fides,  andz  —  -  z:  £  v^^  +  "^ 
2  2  4 

^  extracting  the  root  •>  and  tier  ef  ere,  z~ 


a  a* 

4*  t"  /iwr  ^^  fofitive  value  cf  z,  and  z  ~  —  \/bc  H  -- 

4 

*  ^z 

-f~w  negative  one  -,  forfince  be  +  —  is  greater  than 

~2  >  confluent  fy,   \/bc  +  —  is  greater  than  </—  s 


(     345     ) 

~    P  a 
And  tfartforc,  z~  —  \Sbc  +  —  -f  *£  *•*  always  a  ne 

gative  quantity. 

And  in  like  manner*  the  value  of  z  determined  in  the 

a* 
third  form,  viz.  z*—az  —  —  be,  is  z  =i  ±.\S  —  —  fa 

a  a* 

4—  >    where  both  the  values  of  z  will  be  fejitive,  if  •—  • 

#*"""  a 

i$  greater  than  be  >  for  then  z  —  :  v/  --  ^r'^-N'  '**<[• 

vidently  a  pojitive  quantity  ;  and  in  thejscond  value  of 

a*  a  az 

z,  viz.  z  —  —  \/  —  —  ^  +  "»    ^  V  pl<tin>that  — 

/V  greater  than  —  —  *  bc}fince  —  is  greater  than  Ic  ; 


and  therefore,  the  \S —  /V  greater  than  \/ —  — fa  ; 

a*  a*          a    . 

confequently,  z  —  —  y  —  —  be  +  \S~(~"')tsa 

fofitive  quantity.     But  when  be   is  greater  than  — 

a* 

then  —  —  be  is  a  negative  quantity  5  and  fince  the 

fquare  of  any  quantity  (whether  fofitive  or  negative) 

is  always  fofitive  ;  it  follows,  that  ^  —  —  fa  is  im~ 

pojjibley   or  imaginary  j    and   consequently,     z  ~  ± 

a*  a 

Y/  —  —fa^~  is  imaginary,     therefore,  in  the  third 

form, 


(      346      ) 


a* 
form,  when  be  is  greater  than  —  the  Jolution   tf  the 

equation  will  be,  impojfible. 

EXAMPLES 

Of  determining  the  value  of  the  unknown   quan 
tity  in  quadratic  equations. 

Given,  AT*  -f-  4*^:32,  to  find  the  value  of*-. 

Firft,  AT  -f-  4.x  -f-  4  z:  32  +  4,  by  adding  thefquare 
of  half  the  co-efficient  to  loth  fides  : 

Then,  \/x*  -f-4#-f-  4~  d:  V/3^  : 

That  is,  x  +  i  —  +  6  ;    or,  x  z:  ±  6  —  i  n  4,   cr 

—8:  Either  of  which  fubftituted  for  x,  will  f  reduce 
the  given  equation. 

Given,  3**  — 9*= — 6,  to  find  x, 

Firft,  x*  +- -3^  —  — -zby  dividing  the  whole  by  3  : 

9     9 

>  a:1  —  3oc-f--z:"-  —  2^v completing thefquare : 
4     4 

3  9 

therefore,  x  —  ~  —  +  v^~  —  2^y  extracting 

the  root : 

Given,  ay*  -~  bv  -~  c  ~  d,  to  find  vt 
Firft,  av't—l>vzz4 — c  h  tranjpofition  ; 

„  j ' .  *    J-f 

^fW  ^  —  -  v  =  — — 
a  a 


(      347      ) 

I  b*        d—~C        b"' 

therefore,  v*--  v  +—;  =  —  +  —  fy  com- 

fie  ting  thejquare  : 

b  J^c       ~ 

Whence  y   v  ~  —  ~  ±  v  *  -  -f  —  ?  by  evokt- 


tlon  : 


b         ,d—c       F 
Or,  v=t~±  </•-—  +  —  i  £  tranftofaion. 

AI*L  equations,  wherein  there  are  two  terms  which 
involve  the  unknown  quantity,  whofe  index  in  one 
term,  is  juft  double  its  index  in  the  other,  are  re 
duced  to  equations  of  lower  dimenfions,  in  the  fame 
manner  as  quadratics. 

B 

THUS,  v6  -J-  bv*  IT  d  ;  and  vn  -f-  ^T  =  c>  are  re 
duced  by  completing  the  fquare,  and  extracting  the 
root,  as  in  quadratics  ;  and  the  value  of  the  un 
known  quantity  determined  by  extruding  the  root 
of  the  reiulting  equation  -,  as  in  the  following 

EXAMPLES. 

Given,  *y4  —  2^a  =  224,  to  find  the  value  of  v. 

Firft,  V*  —  2<ya-f-  1  =  224+  i  n  225  by  com- 
fleting  the  fquare  : 

And  v*  —  i  zz  v/22^  ly  evolution  : 
Or,  vz  zz  v^^25  4-  i  by  tranfpofition  : 


Whence^  <u~  1/225  -{-  ijz  ~  4. 

n 

Given,  bvn  +  cv*  —  d  —  e>  to  find  v. 

Firft y  bvn  +  ctf  r=  e  +  d  by  tranfpofiticK, 

c    ^      e  4-  d 
vn  +-V~~  ^—-  ly  divifiw  : 


And  V 
fleting  thejquare  : 


(      348      j 


c  •  "        c1       e 


n         r  £   \    d         c*" 

therefore,  <u*  -f       =  ±  v^-*  + 


TT71  S€   ~^    d  C*  C 

Whence,  v  =  4-  v/-^—  -f  ~TT—  T 

r      i 


CHAP.     XVII. 

f  he  SOL  UriO  Nofa  Variety  of  $UE  S  7*70  N$, 
Producing  QUADRATIC  EQUATIONS. 

i.  TT  7  H  AT  two  numbers  are  thofe,  whofe  fum 
\\    is  20,  and  their  product  96  ? 

Call  one  of  the  numbers  w  -3  then  will  20  —  w  be  the, 
ether  : 

And  w  x  20 —  w  z:  $6  by  the-queftion  : 
That  is,  low  —*w*  —96  : 
Or,  *ya  —  2Ow  rr  —  96  ^y  tranjfofition  : 
And  wz  —  2oze>  +  ioo  ~  100 —  96  ^y  complet 
ing  thefquare  : 

Therefore,  <oo  —  10  =:  Hh  \/ioo — -96  =  ±  v/  4 
^  di  2  ^y  evolution  : 

Ory  w~  +  2  +  10=  I  a  or  8,  andia — -w  rz  20 
<— -  1 2  ~  8  tbe  oiber  number* 

i.  What  two  numbers  are  thofe,  \vhofe  fum  is  36, 
and  the  fum  of  their  fquares  720  ? 
Put  w  for  the  greater  number  : 
Then  will  36  —  w~tke  other  :  * •  , 


(      349      ) 

And  w*  «/-  36  —  w]*  —  7  20  £y  /£*  quefiion  : 
Tfttf/  /V,  wa  +  1  296  —  7  2W  -f-  w2  ±:  7  20  : 
Or,  2w*  —72-12;  =  —  576  £y  tranfpofition  : 
dndw*  —  36  w  ~  —  188  by  dhifion  : 
Wherefore,  w*  —  $6w  4-  324  n:  324  —  288  iz  3^ 

by  completing  the  f  quart  : 

Consequently,  w  —  i8ir  ±  \/  36  z:  6  by  evolution: 
Or,   w  —  6  -f-  1  8  —  24,   £?/^/  36  —  •  w  —  36  —  24 

n  12  : 

3.  What  number  being  divided  by  the  product 
of  its  two  digits,  the  quotient  will  be  2  j  and  if  27 
be  added  to  it,  the  digits  will  be  inverted  ? 

Put  w  and  y  for  the  two  digits  : 
Then  will  10  w  -f  y  be  the  number  fought,  by  the 
nature  of  notation  : 


~~wj~       >2 
And  low+y  +  27  ~  iqy  -{-  w 

Or,  9  w  =  97  —  27  ^y  tranfpofition  : 

QV—  -  27 


1 

\bytb 
] 


9 

IO-K;  +  y  -:  207  ;  whence,  (by  writing  for 

its  equal  y  —  3,  in  the  equation  low  +  y  rr 
£<?/  .1  qy  —  30  +  y  =  2^4  —  6y  : 


—  2y2  =  305  or,  2y4  —  I7jn  —  30   by 
tranfpojition  : 

Whence,  y*  —%l.y  =  —  15  ^y  dhifion  : 

289       289  49 

Andy*—  ^y^-^--j._ls-^by  cm. 

Dieting  thefquare  : 

Or,   >  —  —  =  ±  \/-7  =  -  by  evolution: 
4  10      4  • 

Y  y 


Confejumtly,  J- 


(      35°      ) 


Therefore  36  zV  /&*  number  required. 

4.  To  find  three  numbers  in  geometrical  propor 
tion  continued,  v/hofe  fum  is  78  ;  and  ifthefum 
of  the  extremes  be  multiplied  with  the-mean,  the 
product  will  be  1080. 

Putv  ~  leaft  extreme  >  andz  the  greiter  ;  alfo>y  — 
mean  : 

*Ihen  will  v  -f-  y  -f-  z  zz  7  8  7   r      r          «• 

-  -    J  >  fa  the  queftion. 

And  v  +  z  X^=  1080  i 

W^/  is,  vy  -f-  zy^z  io8Oi  andvy  + 
3y  wultif  lying  the  fir  fl  equation  ivith  y  ; 

Whence^y%i^.  (by  writing  for  vy-\-zy  it 
787  —  1080, 

Or3  j2  —  78yn—  1080: 

Andy*  —  78^4*15213:1521  —  108011441 
completing  thejquare  : 

And  therefore,  y~*  39  rz  ±  y/44I  ~  ± 
ticn  : 

Or,  y~  39  4;  21  =  (becaufe  39  -J-  21  zz  60,  /V 
greater  than  thejum  of  the  extremes^  which  is  abjurd) 
39—  21  —  18  : 

But>vz  ~y*  =  324  ^j  the  nature  of  the  proportion  : 


)  v^.  -  >    which  wrote  for  <v  in  the  e- 


quation  vy  +  zy  zz  1080,  JT^J  -  --  1-  zy  rr  1080  : 


,  5932  -f-  iSs;1  zz  10802  : 
Or,i  8z*  —  10802;  zz  —  5932  by  tranffofition  : 
And  zz  —  602  zz  —  324  by  divifion  : 
Therefore^  z*  —  602.^-900  —  900—324= 
completing  tkejq 


(     35'      ) 

Whence,  z  —  30  :=  i  \/  5  7  6  zr  +  24  £y  evolution : 
Or,  z  zz  30 -f-  24=54,  and  ?;  zz  78  —  2  —^  =  78 

—  54  —  1 8  zz  6.     Iberefore,  6,  18,  0»//  54,  are  tbe 

numbers  required. 

5.  There  are  three  numbers  in  geometrical  pro- 
grefllon,  whofe  fum  is  117,  and  the  fum  01  their 
fquares  7371  :  What  are  thofe  numbers  ? 

Call  the  numbers  x}y  and  v  : 

Then  will  #  +  y  -f*  i;  zz  1 17  ?   7 

And  x*  +  /+.T-  =  7,71  5  ^ 

A'i;  —  jy*  ^y  /^^  nature  of  the  proportion  : 
A;  4-  ^  zz  117  — j  by  tbe  firft  equation  : 

Whence^   x*  4-  2^1?  -f*  i;z  =^=  13689  —  234^ 
4y  involution : 

Buty  ixv  zr  2jz,  which Julftituted  for  2.x<v  in  the 
la  ft  equation,  gives  #*  +iy  -f'y2—  13689  —  234^  -J* 
/*J? 

Or,  ^a  +<ya  zr  13689  —  23/y — ^*  : 

^^A?1  -^  «y*  =7371  — ^*  ^y  thejecond  equations 

Confequently,  737  i  — jy*  zi  13689 —  234^  —  >a  5 

Or,  2347  =  13689—7371  —6310: 

6310 
Whence^y—— — •  zr  27,  and xv—y*  n  729  : 

OT" 

7  ^9 
Or,  ^  ^ ,  whichfubftituted  in  ihs  equation  x+ 

729  729 

y+v  =  i  IT  .gives  —+27  +  i?=n7i    cr,    ~ 

+  T;—  117— -27  —  9°  : 

Whence y  729  -j-.i;*  ~  901;  ^)f  multiplication  : 

Or,  v*  —  901;  zz  — 729  by  tranJpofit'iQn  : 

And  therefore,,    v*  —  901;  •+  2025  zz  2025  —  729 

-zz  1296  ^  completing  tb?f%ua,re  : 


Conjequently,  v  —  45  —  -f  \/  1  296  =  36  £y  evolu 
tion  ; 

729 
Or,  ^  —  45  +  36  —  81,  andx—  —  -  ~  9. 

And  the  number  3  required,  are  9,  27,  81. 

MISCELLANEOUS      QUESflONS, 
their   SOLUTIONS. 


I.  Suppofe  two  cities,  A  and  B,  whofe  diftance 
from  each  other  is  216  miles  -,  and  that  two  cou 
riers  fet  out  at  the  fame  time,  one  from  A,  and  the 
other  from  B;  the  firft  travels  10  miles  a  day,  and 
the  other  4  miles  lefs  than  the  number  of  days  in 
which  they  will  meet.  Query  the  number  of  days 
before  they  meet  ? 

Put  x  ~  number  of  days  required  : 

Then  will  i  o#  -J-  x  —  4  X  #H  2  1  6  by  the  queftion  : 

That  isy   IQX  +  #*  —  ^x  —  zib  -t  or,  x7"  4-  6x  — 


And  x*  +  6^  +  9z:2i64-9^i:  225  : 
Whence,  A;  4-  3  n  ^  v/  2  25  n:  1  5  s  cr,  Arn  1  5  —  ^  3 
rr  12,  the  number  of  days  required. 

2.  A  traveller  fets  out  from  the  city  A,  and  tra 
vels  at  the  rate  of  9  miles  an  hour  ;  and  another  at 
the  fame  time  fets  out  from  the  fame  city,  and  fol 
lows  him,  travelling  the  firft  hour  4  milts  5  the  fe- 
cond  5  j  the  third  6,  and  fo  on,  in  arithmetical  pro- 
greffibn  :  In  what  time  will  he  overtake  the  firil  ? 

Put  x  —  number  of  hours  in  which  the  firft  uill  be 
Overtaken  : 

f  hen  will  $x  rr  the.  diftance  he  travels  : 

x—  i  X  I  +4*4—  #+7  • 

And 


(      353      ) 

~ —  ^  dtflance    tbe    other 

travels  before  he  overtakes  the  firft,  by  (be  nature  of  the 

x*  -f-  jx 
proportion  :  Confequently,  zrcj*  by  the  quef* 

tion : 

Or,  x*  -4-  7*-^  i8# :  Whence,  #  +  7  =  18  5   0r,  * 

• — ;  1 1  hours,  the  time  required. 

3.  There  arc  four  numbers  in  geometrical  pro- 
greffion,  the  fum  of  the  extremes  is  84,  and  the  furn 
of  the  means  36  :  What  are  thofc  numbers  ? 

Put  v  andy  for  the  means  : 


will  —  and  —  be  tbe  extremes  by  tbe  nature  $/ 

tbe  proportion  : 

V*       JV*  I  ly  tbe  queftion  : 


H*ti*t  p  for  vy) 

p  b. 

But,  v3  -i-jy3  =  (by  problem  1  8 


Confidently,  pb  ~  a*  —  $ap  ;  cr*  p  ~  T  -  -  ^r 

^jy  fubjlitution  : 

Therefore,  V*  -}-j3  r:£f;  e?r,  -j;x  s=^f  —  jyj  : 
But,    v  ~  tf  —  y  ;  therefore,  v*  ~  a*  —  ^a*y  •+« 


And  tberefwe3  y*  ~~  cy  H  —  -  -  " 

*a  And 


And-g*  — *  ay  rt~  7-  — 


^w^,7— -SAifrr 


, 

-  —  27,  «»rfv  —  36—- j,r:9;  therefore^  — 


nnmlers  re 
quired. 

4.  Suppofe  two  cities,  A  and  B,  whole  diflance 
from  each  otiier  is  152  miles  j  and  that  two  men  fee 
out  at  the  fame  time  from  thole  cities  to  meet  each 
dtrkr  ;  the  one  which  goes  from  A,  travels  the  firft 
day  i  mile,  the  fecond  day  2,  the  third  day  3,  and  fo 
on;  and  the  one  wnich  lets  out  from  B/  goes  the 
firir.  day  4  .miles,  the  fecood  day  7,  and  the  third  io> 
a^d  fo  on.  Qjery  the'  number  of  days  before  -they 
nicer,  ,  and  the  .number  of,milts  that  each  travels? 

Put  y  ~  number  of  days  before  ttiey  meet  ; 

•        15?  =  ,  5  3  ij,  the  gueJHe*  : 


Ay  ^  -\~  6  y 
!  That  iis>      •         '  *=r  152  ;  tf>,*4j*  -f  6y  —304  : 


^j*  +- j-f  ~6rr  76 

122 

*  y  ^  4  ~^  —  ^ .  16 

8.  ,    "    ' 

%   \ 


V  *    —!••  V 

yl- ;  —-36,  tks   number-   cf  miles 

travelled   by  tbe   ont   which  Jat  -out  frcm    A>    and 

i 

ZT  1 1 6,  the  dljtancs  travelled  by  the  other. 


C  H  A  P.      XVIII. 

Oftbs  GENESIS,  cr  'FG&MATION  of  E- 
^U  AT  IONS  in   GENERAL. 

ALL  equations  of  fuperior  order,  are  confider- 
ed,  as  produced  by  the  multiplication  of  equa 
tions  of  inferior  orders,   that  involve  the  fame  un 
known  quantity. 

Thus,  a  quadratic  equation  may  be  cdnndered  as 
generated  by  the  multiplication  of  two  fimpte  equa 
tions  ;  a  cubic  equation  by  the  multiplication  of 
three  flmple  equations,  or  one  qiuiciratk  and  one 
fimple  equation  ;  and  a  biquadratic  equation  by  the 
multiplication  of  four  fi.nple  equations,  or  two  quad 
ratic  equations,  or  one  cubic  and  one  fimple  equa 
tion. 

Suppofe  w  to  be  the  unknown  quantity,  and  a,  4 
ct  d>  &c.  its  feveral  values  in  any  fimple  equation  ^ 

That  is,  w~a3  w  ~  ^,  w  n  c,  w~d,  dec.  Thta 
by  tranfpoiition,  w  — a  ~  o,  w  —  b  no,  w  —  rzrr 
o,  w  —  d  ~ o,  &c.  And  the  produdl  of  two  of  thefe 
equations  as  w  — a  x  ^  —  ^zrzO,  gives  a  quadratic 
.equation,  or  one  of  two  dimenfions. 

The  produft  of  any  three  ;  as  w  —  a  X  w  —  b  X 
w  —  c  r-  o,  produces  a  cubic  equation,  or  one  of 
three  dimenfions,  The 


(     356     ) 

The  pro&3&  of  any  four  of  thertt  5  3$  w  —  ^  X 
•se?  — £  x  '^  —  c  X  w  —  d  zr  o,  produces  a  biquad 
ratic  equation,  or  one  of  four  dimenfions. 

Hence  it  appears,  that  in  every  equation,  the 
higeft  dimenfion  of  the  unknown  quantity,  is  equal 
to  the  number  of  fimple  equations;  that  generate  that 
equation  -,  and  therefore  it  follows,  that  every  equa 
tion  has  as  many  roots,  or  values  of  the  unknown 
quantity,  as  there  are  units  in  the  higheft  dimenfion 
of  that  unknown  qnantitv.  For  fuppoie  an  equation 

-*.  w  —  a  X  "Jo  —  b  X  w  —  f  —  o  ;  and  that  for  iff 
you  fubflitute  any  of  its  values  (#,  b  or  c)  in  the  giv 
en  equation,  then  all  the  terms  of  an  equation  will 

vanifh  ;  for  if  w  ~a,w=.b,  and  w  —  r,  then  w  —  a 
X  ^  —  b  X  w  —  <:  ~  o,  becaufe  each  of  the  factors 
are  equal  to  nothing.  And  after  the  fame  manner, 
it  appears 3  that  there  are  three  fuppofitions  that  give 

ce;  —  a  x  w  —  ^  X  w  ~~  c  ^  °  :  ^ut  fi nee  there  are 
no  other  quantities  befides  thefe  <*,  £,  r,  which  fub*-' 

ftituted  for  w-in  the  equation  w —  a  X  w. —  £  X 
w  — c  n:  o,  will  make  all  the  terms  vanifh  ;  it  fol- 

lows,that  the  equations  —  a  X  w  —  ^  X  w  —  c~ 
o,  can  have  no  more  than  thefe  three  roots,  or  ad 
mit  of  more  than  three  folutions.  For  if  you  fubfti- 
tute  for  is;  in  the  propofed  equation,  any  other  quan 
tity  ^,  which  is  neither  equal  to  a,  £,  nor  c  ;  then 
neither  e  —  a,  e — b>  e  —  cy  is  equal  to  nothing; 
arid  confequently  their  product  e—a  X  e  —  b  X  e~cy 
cannot  be  equal  to  nothing,  but  muft  be  fome  real 
product :  So  that  no  other  quantity,  btfide's  one  of 
thofe  before-mentioned,  will  give  a  true  value  of  w 
in  the  propofed  equation.  And  therefore,  no  equa 
tion  can  have  more  roots  than  it  contains  dimen 
of  the  unknown  quantity.  To 


(     351     ) 

To  be  more  plain  :  Suppofe  that  AT*—  icx*  +  35** 
•—  50*  +  24  mo,  is  the  equation  to  be  rtfolved  ; 
and  that  you  find  it  to  be  the  fame  as  the  produdl  of 

#~ —  i  X  x —  2  X  #  —  3  X  #  —  4 :  Then  you  will 
infer,  that  the  four  roots  or  values  of  x,  are  i,  2, 
3,  and  4  j  for  any  of  thefe  numbers  fubftituted  for 
A:,  will  make  that  product,  and  confequentiy,  .v4-— 
IOAT3  -f-  35*v*  —  5O#-{-  24  equal  to  nothing,  accord 
ing  to  the  propofed  equation. 

THE  roots  of  equations  are  either  pofitiye.or  ne 
gative,  according  as  the  roots  or  values  of  the  un 
known  quantity  in  the  fimple  equations  which  prod 
uce  them,  are  pofitive  or  negative.  Thus,  if^z:—  #, 
vn  —  b,v=.  —  r,  viz — d\  then  will  v  +  ^n:o, 
i)  -\-b  —  o,  *v  -\-  c  ~  Q,  and  v  +  d  ~  o ;  and  con-* 

fequently,  v  +  a  X.^  +  ^.X^  +  ^X  ^  •+-  d  zi  o, 
will  be  an  equation  whofe  roots  —  ay  —  ^,  —  r,  — ^/, 
are  all  negative.  And  after  the  fame  manner,  if 
v  ~  ^,  ^  zr  —  £,  -i;  —  f ,  the  equation  v  —  a  X  ^HhJ 
X  ^  —  f,  will  have  its  roots  -f  *,  —  £,  -}-  r. 

BUT  to  difcover  when  the  roots  of  an  equation  are 
pofitive,  and  when  negative,  and  how  many  there  are 
of  each  kind,  it  will  be  neceflary  to  confider  the  figns" 
and  co-efficients  of  equations,  generated  from  the 
multiplication  of  thofe  fimple  equations  that  produce 
them;  which  will  be  beft  underftood  by  confidcring 
the  following  table,  where  the  fimple  equations  v— ^ 
V  —  by  v  —  f,  &c.  are  multiplied  continually  with 
one  another,  and  produce  fuccefliv.ely  the  higher  e- 
quations. 


— a 


v  —  a 
X  v  —  b 


n:^*  —  avl          »  ,. 
^    >  +  ab  —  o,  a  quadratic 


vz  —a  1 

—  b    S  X  V1  +  ab  I 

—  c  J  -f  ^r  >  x  ^  —  abc  =:  o,  4  ^ 

-f  ^  J  '[equati 


cubic 
equation 


o, «  biquad* 


FROM  the  infpeflion  of  thefe  equations  it  appears 
that  the  co-efficient  of  the  firft  term  is  unity  or  i. 

THE  co-efficient  of  thefecond  term,  is  the  fum  of 
all  the  roots  (ay  b>  c,  d}  with  contrary  figns. 

THE  co-efficient  of  the  third  term,  is  the  fum  of 
all  the  produces  of  thofe  roots  that  can  poflibly  be 
made  by  multiplying  any  two  of  them  together. 

THE  co-efficient  of  the  fourth  term,  is  the  fum  of 
all  the  produces  of  the  roots  that  can  be  made  by 

combining 


(     359      ) 

combining  them,  three  and  three  :  And  fo  on  for 
any  other  co-efficient.  The  lad  term  is  always  the 
product  of  all  the  roots,  having  their  figns  changed. 

NOTWITHSTANDING  thofe  fimple  equations  made 
ufe  of  in  the  foregoing  table,  in  forming  the  higher  e- 
quations,  are  fuch  as  have  pofitive  roots  ;  yet  the  fame 
reafoning  holds,whether  the  roots  are  pofitive  or  neg 
ative.  Whence,  if  v* — pv*  4-  qv*  —  rv-{-s~  o, 
reprefents  a  biquadratic  equation  ;  then  will  p  be  the 
fuin  of  all  the  roots,  q  the  fum  of  all  the  products 
made  by  multiplying  any  two  of  them  together,  r 
the  fum  of  all  the  products  made  by  multiplying  any 
three  of  them  together,  and  s  the  product  of  all  four* 

IT  likewife  appears  from  infpection,  that  the  figns 
of  the  terms  in  any  equation  in  the  foregoing  table4 
are  alternately  4-  and  —  :  The  firft  term  is  always 
fome  pure  power  of  v,  and  is  pofitive  :  The  fecond 
term  is  fome  power  of  v,  multiplied  with  the  quan 
tities,  —  ay  — £,  — r,  &c.  and  fince  thefe  quanti 
ties  are  all  negative,  it  follows,  that  the  fecond  term 
muft  alfo  be  negative.  The  third  term  hath  for  its 
co-efficient  the  product  of  any  two  of  thefe  quanti 
ties,  ( — #,  •— ^£,  —  c ,  &V.)  and  fince  —  X  —  gives 
•4-  ;  it  follows,  that  the  third  term  mud  be  pofitive. 
For  the  fame  reafon,  the  co- efficient  of  the  fourth 
term,  which  is  formed  of  the  products  of  any  three 
of  thefe  negative  quantities,  muft  be  negative  alfo, 
and  the  co-efficient  of  the  fifth  term  pofitive.  But 
in  this  cafe,  v  —  <?,  v  =  b,  v  —  c ,  v  z:  dy  &c.  that 
is,  the  roots  are  all  pofitive  :  Confequently,  when  the 
roots  of  an  equation  are  all  pofitive,  the  figns  of  the 
terms  are  +  anc^  —  alternately.  But,  when  the 
roots  are  all  negative  \  that  is,  v  —  — a>  v  ~  —  by 
•u  rz  —  cy  v  z=  —  d,  &c.  then  -y-j-^X^-f-^X 
v  4-  c  X  v-J-^=o,  will  exprefs  the  equation  pro 
duced 


duced,  whofe  terms  are  evidently  all  pofitive.  And 
therefore  when  the  roots  of  an  equation  are  all  nega 
tive,  there  will  be  no  change  in  the  figns  of  the  terms, 
tonfequently,  there  will  be  as  many  pofitive  roots  in 
an  equation,  as  there  are  changes  in  the  figns  of  the 
terms  of  that  equation,  and  the  reft  of  the  roots  will 
be  negative. 

HENCE  it  follows,  that  the  roots  of  a  quadratic  e- 
cjuatipn  may  be  both  negative,  or  both  pofitive,  or 
one  negative  and  the  other  pofitive.  Thus,  in  the 

equations*  —  a  1  7       •    •  v  -  -,\ 

L   r  X^  4-  ab  ~    (v  —  a  X  v  —  &4 

o,  there  are  two  changes  of  the  figns,  viz.  the  firft 
term  is  pofitive,  the  fec<5nd  negative,  and  the  third 
pofitive  j  confequently,  the  roots  are  both  pofitive. 

BUT  in  the  equations*  -f-  al  ,  ,       7  —  — 

-f-  b\  Xv  +  a*  =  (v  +  a 

X  v  -f  £)  o,  there  are  no  change  in  the  figns,  and 
therefore  both  the  roots  are  negative. 

AND  in  like  manner,  in  theequation  v*  -\-  a 


_     _ 

ri)  —  ab  IT  (v  +  a  X  "J  —  ^)  o,  one  of  the  roots 
will  be  pofitive,  and  the  other  negative  ;  for  fince 
the  firft  term  is  pofitive,  and  the  lalt  negative,  it  is 
plain,  there  can  be  but  one  change  in  the  figns,  whe 
ther  the  fecond  term  is  pofitive  or  negative. 

HENCE  alfo  it  appears,  how  that  a  cubic  equation 
may  have  all  its  roots  pofitive,  or  all  negative,  or 
two  pofitive  and  one  negative  $  or  two  negative  and 
one  pofitive.  For  fuppofe  the  cubic  equation  is 


a  I 

*    \  X  v*  +"ab  ] 

c  J  +  ac   \ 

+  :lc  J 


X  v  ~  ale  ~  (v  —  a  X 


(      36 1      ) 

^  —  b  X  t>  —  *•)  o,  wherein  there  are  three  changes 
in  the  figns  \  and  confequently  all  three  of  the  roots 
pofitive. 

AGAIN,  fuppofe  the  cubic  equation  is  of  this  form, 


v3  —a  *} 
—  b    \  X  v 

+   *    J 


X  v  +  ale  =     v  — «  X 


*°  — I  X  v  —  0  °j  where  there  are  two  changes  in 
the  figns  -,  for  if «  -}-  b  is  greater  than  r,  then  the  fe- 

cond  co-efficient  — a  —  b  -J-  f  muft  be  negative  j  if 
a  +  b  is  lefs  than  r,  then  the  third  term  will  be  neg 
ative  5  for  its  co-efficient  ab — ac  —  bc{~'ab—t 
X  a  +  b)  is,  in  this  cafe  negative,  becaufe  the  prod 
uct  a  x  b  is  always  lefs  than  the  fquare  a-\*b  X  «+£> 
and  confequently,  much  lefs  than  f  x  *  -f-  ^  5  and 
fince  tnere  cannot  be  three  changes  in  the  figns,  the 
firft  and  lad  terms  having  the  fame  fign  ;  it  follows, 
that  two  of  the  roots  of  the  propofcd  equation  are 
pofitive,  and  the  other  negative. 

IN  like  manner,  the  equation  v3  +  a  -\- b  —  "ry* 
H-  ab  —  ac  -~-  bcv — ^^r~o,will  have  two  of  its  roots 
negative,  and  the  other  pofitive  j  for  if  a  -|-  b  is  lefs 
tha».  cy  the  feco'nd  and  third  terms  muft  be  negative, 
by  what  was  proved  in  the  laft  example  -9  and  if  the 
fecond  term  is  pofitive,  that  is,  a  -f-  b  is  greater  than 
f,  it  is  plain  there  can  be  but  one  change  in  the  figns, 
and  confequently  but  one  pofitive  root,  the  other 
two  being  negative. 

AND  by  parity  of  reaion,  the  pofitive  and  negative 
roots  of  the  other  equations  may  be  difcovered  ;  . 

this 


c_£f!_) 

this  method  being  general,  and  extends  to  all  kinds 
of  equations  whatever. 


CHAP.      XIX. 

CONCERNING  the  TRANSFORMATION 
cf  EQUATIONS,  and  EXTERMINAT 
ING  their  INTERMEDIATE  TERMS. 


N  Y  equation  may  be  transformed  into  another, 
^  ^  whofc  roots  fhall  be  greater,  or  lefs  than  the 
roots  of  the  propofed  equation  by  any  given  differ 
ence  (?)  by  the  following 


A 


RULE. 


ASSUME  a  new  unknown  quantity  (j)  and  conneft 
it  with  the  given  difference  (e}9  with  the  fign  +  or 
~,  according  as  the  roots  of  the  propofed  equation 
are  to  be  increafed,  or  diminifhed  ;  and  make  this 
aggregate  equal  to  the  unknown  quantity  (#)  in  the 
propofed  equation ;  then  inilead  of  the  unknown 
quantity  (#)  and  its  powers  in  the  propofed  equation, 
fubilitute  this  aggregate,  (y  ±  e)  and  its  powers; 
and  there  will  arife  a  new  equation,  whofe  roots  will 
be  greater  or  lefs  than  the  roots  of  the  propofed  e- 
quation,  as  required, 

EXAMPLES. 

i.  Let  x3 — px*+qx  —  riro,  be  an  equation 
to  be  transformed  into  another  whofe  roots  fhall  be 
lefs  than  the  roots  of  the  propofrd  equation,  by  the 
difference  e,  AJJume 


dflume  x  ~y  +  e: 
Then  will  x*  —  j3  -f-jyv-f  y^  4.  ^ 


r     = 


qe 

*     1  ^nation  requir. 


2.  Let  #x  —  1  1  #  -|-  30  n  o,  be  transformed  into 
an  equation  that  fhall  have  its  roots  lefs  than  the  roots 
of  the  propofed  equation  by  the  difference  4. 

AJfume  x  ~  y  -f-  4  : 

16  : 


—  n^zz      —  iij—  44 

+  3°  =  +3° 


J*  —  j^  +  2—O,  w  /^  equation  required. 

IN  the  firft  example  of  the  foregoing  transforma 
tions,  the  co-efficient  of  the  fecond  term  in  the  tranf- 
formed  equation,  is  3*  —  p  ;  and  if  you  fuppofe  £~ 
-J./>,  and  therefore,  3*  —  /?  —  o  j  then  the  fecond  term 
of  the  transformed  equation  will  vanifh.  Let  the 
propofed  equation  be  of  #dimenfions,  and  the  co-ef 
ficient  of  the  fecond  term  —  f  ;  and  fuppofe  # 

P 
=j  +  -*  then  if  this  value  be  fubftituted  for  ^  in 

the  propofed  equation,  there  will  arife  a  new  equa 
tion  that  fhall  want  the  fecond  term.  For  if  p  = 
fum  of  all  the  roots  of  the  propofed  equation,  and  x 

p 
~y  -f.  -  ;  it  follows,  that  each  value  of?  in  the  new 

equation,  will  be  lefs  than  the  value  of  x  in  the  pro- 

P 
pofed  equation,  by  ->  and  fince  the  number  of  roots 

is  »,  it  follows,  that  the  fum  of  the  values  of  y>  will 

be 


be  lefs  than  p,  the  fum  of  the  values  of  %,   b#  n  x  ± 

~  ^  ;  that  is,  the  fum  of  the  values  of  j,  is  +/>  —  p 
^  o  ;  and  fince  the  co-efficient  of  the  fecorid  term 
in  the  equation  ofy,  is  the  fum  of  the  values  ofj, 
viz.  -f-  p  —  p3  which  is  equal  to  nothing;  it  follows, 
that  in  the  equation  of  j,  arifing  from  the  fuppofi- 

tion  of  Af—jy  -{-  ~,  the  fecond  term  onuft  vanifh  : 

And  therefore  the  fecond  term  of  any  equation  may 
be  exterminated  by  the  following 

RULE. 

DIVIDE  the  co-efficient  of  the  fecond  term  of  the 
propofed  equation  by  the  index  of  the  higheft  power 
of  the  unknown  quantity;  and  a  flu  me  a  new  un 
known  quantity  (  y  )  and  annex  to  it  the  faid  quotient 
with  its  fi-gn  changed  ;  then  put  this  aggregate  e- 
qual  to  the  unknown  quantity  (x}  in  the  propofed 
equation,  and  inftead  of*  and  its  powers,  write  thfs 
aggregate  and  its  powers,  and  the  equation  that 
arifes  fhall  want  the  fecond  term. 

EXAMPLES. 

Let  the  .equation  x*  —  8  #-J-  i.a  ~  o,  .be  propo- 
ied  to  have  its  fecond  term  exterminated. 
8~2~—  4; 
,  x~y-\-4.,  per  rule  : 
Then,  x*  —y*  +  8j  +  16 
—  8^~      —  8jx  —  32 


4  =  0 

HENCE, 


(     36$      ) 

HENCE  it  appears,  that  a  quadratic  equation 
be  refolved  without  completing  the  fquare,  by  ex 
terminating  the  fecond  term  ;  for  fincejy1—  4  —  o  ; 
or,y*r:4,  and  jy  ~  v'  4j  we  fhall  have  x  —y  +  4:=: 


Let  the  fecond  term  of  the  equation  #3 
—  34  —  o,  be  exterminated. 


Then,  x3  ~y3  -f-  QJ*  +  27^  +  27 
_9#*n  —9?*  —  547  —  81 
+  26%  —  -j-26y-f-78 

—  J4^  ___  _—  34 

y3        *      —  jx  —  10  =  0. 

WHEN  the  fecond  term  in  any  equation  is  want 
ing,  it  is  plain,  that  the  equation  hath  both  pofitive 
and  negative  roots  ;  and  fmce  the  co-efficient  of  the 
fecond  term  in  any  equation,  is  the  difference  be 
tween  the  fum  of  the  pofitive,  and  fum  of  the  nega 
tive^  roots  j  it  follows  therefore,  that  when  the  pofi 
tive  and  negative  roots  are  made  equal  to  each  other, 
that  difference  vanifhes.  Confequently,  v/hen  an  e- 
quation  has  the  fecond  term  wanting,  the  fum  of  the 
pofitive  roots  is  equal  to  the  fum  of  the  negative 
ones. 

HENCE,  by  the  foregoing  transformation  of  equa 
tions  and  the  exterminating  their  fecond  terms,  the 
pofitive  and  negative  roots  are  reduced  to  an  equal 
ity,  and  the  folution  of  the  equation  thereby  render 
ed  more  eafy. 

IF  the  equation  <v*  —  pv*  -\-qv  —  mo,  be  trans 

formed  into  another,  by  affuming  <v  —  y  -}-<?,  the 

co-efficient  of  the  third  term  of  the  transformed  e- 

quation  will  be  y*  —  ipe  +  q-,  now  if  we  fuppofe  this 

Aaa  co-  efficient 


(      366      ) 

co-efficient  equal  to  nothmgV  aftd  refolve  the  quad- 


ratlc  3**  — r 2p 4-  j  2=  o  we  Ihall  have.*  g:?...-; ;. ,-.  .:'V  ~ 

»3 

which  fubftttuted  for  *  in  the  equation  v~y-+-*9 
the  third  term  of  the  transformed  equation  will  van- 
Ifb  :  Alfo,  if  the  pfopofed  equation  be  of  n  dimen- 
fions,  the  value  of  e,  by  which  the  third  term  is  to 
TDC  exterminated,  is  found  by  refolving  the  quadra- 

IP                    CLO  -\ 

tic  equation  <?* -\-~±~ X  e  +      .    7 z:  o,    tnat  is, 

n  n  Xn—  i 

by  finding  the  value  of  e  in  the  co-efficient  of  the 
third  term  of  the  transformed  "equation,  when  that 
co-efficient  is  equal  to  nothing.  And  in  like  man 
ner,  the  fourth  term  of  any  equation  may  be,  exter 
minated,  by  foiving  a  cubic  equation,  which  is  the 
co-efficient  of  the  fourth  term  of  a  transformed  equa 
tion  :  And  after  the  fame  manner,  the  other  terms 
may  be  taken  away. 

THERE  are  other  transformations  which  are  of  ufe 
in  the  refolution  of  equations;  of  which  the  moft 
ufeful,  and  the  only  one  that  we  fhall  confider,  is, 
when  the  higheft  term  of  the  unknown  quantity  is 
multiplied  with  fome  given  quantity,  to  transform 
the  equation  into  another  that  ftiall  have  the  co-effi 
cient  of  the  higheft  term  unity. 

LET  the  propofed  equation  be  atf  — pv*  -f-  ^i;  — 
rzz  o ;  and  fuppofe  av  ~yy  then  v  ~y  -i-  ay  and 
this  value  fubftituted  for  v  in  the  propofed  equation, 

there  will  arife  -^L-  —  -EL-  -f  S2L  — r  ir  cyx>r  — 
a1  a*  a  a* 


—  ?2L  -|_  zL  —  r  n  o,    and  by   multiplying   the 
a*        'a 

whole  by  a*,  we  fhall  have  y*  — £yl  -\-qay  —  ra* 
r:O>  which  gives  the  following 

RULE. 


wfob     RULE.  - 



CBANGE  the  unknown  quantity  (_v)  ii>  the  propof 
ed  equation,  into  another  (y)r  prefix  no  .co-efficienC 
to  the  firft  term,  pafs  the  fecond,  multiply  the  third 
-term  with  the  co- efficient,  of  the  higheft  term  of  the 
unknown  quantity  iathe  propofed  equation,  and  the 
fourth  term  by  the  fquare  of  that  co-efficient,  the 
fifth  by  the  cube;  and  fo  on,  and  the  higheft  terra 
of  the  unknown  quantity  in  the  refulting  equation 
(hall  have  its  co-efficient  unity,  as  required. 

EXAMPLES. 

Let  the  equation  iv*  +  6v— 36  zz  o,  be  changed 
into  another  that  will  have  unity  for  the  co-efficient 
of  the  higheft  term  of  the  unknown  quantity. 

Tbus,yz  +  6y  —  36  X  2  =i  o ;  or> y*  4-  6y  —  72 
r:o,  is  the  equation  required. 

The  finding  the  roots  of  the  propofed  equation, 
and  all  others  of  the  like  kind,  will  be  very  eafy  when 
the  roots  of  the  transformed  equation  are  found -\  fmce 
1}  =  (in  this  cafe)  ~y. 

Transform  the  equation  5i>3—  101;*  H- -i6v  — 93 
m  o,  into  another  that  the  higheft  term  of  the  im- 
known  quantity  may  have  an  unit  for  its  co-efficient. 

Tbusyy*  — IQJ  4-  8qy  —  2325  =  o,  is  the.  equa 
tion  required. 


CHAP. 


(      368      ) 


CH  A  P.     XX. 

Of  the  RESOLUTION  of 

by  DIVISORS. 

IF  the  laft  term  of  an  equation  is  the  produft  of 
all  its  roots  ;  it  follows,  that  the  roots  of  an  e- 
quation  when  commenfurable,  will  be  found  among 
the  divifors  of  the  laft  term  ;  which  gives  the  fol 
lowing 

R  U L  E, 

TRANSPOSE  all  the  terms  to  one  fide  of  the  eqtia- 
tion.  Find  all  the  divifors  of  the  laft  term,  and 
fubftitute  them  fucceflively  for  the  unknown  quan 
tity  in  the  propofed  equation  $  and  that  divifor, 
which  fubftituted  as  aforefaid,  gives  the  refult  zi  Q, 
is  one  of  the  roots  of  the  equation.  But  if  none  of 
the  divifors  fucceed,  the  roots  of  the  equation  are 
for  the  moft  part,  either  irrational  or  impoffible.' 

Note.  If  the  laft  term  cf  the  fropofed  equation  is 
large,  and  confequently  Its  divifors  numerous  ± 
they  may  be  diminijhedy  by  transforming  the  equa 
tion  into  another,  by  the  rules  of  the  laft  chapter. 

EXAMPLES. 

Find  the  roots  of  the  equation  x3  —  4**  -j-  io#— 
j  2  =  o. 

Here  the  diyifors  of  the  la/I  termy  are  .  i,  2,  3,  4,  6, 
, —i,  — 2,   — 3,  — -  4,  —  6,  —  12,   which  Jub- 
for.x*  ..:........<>      ~ 

Gives, 


f  1-^44-  io—i2  ~— -5 
j    8  —  1 6 +  20  —  1 2  =  o 
j    Gives*   <{    27  —  36  +  30  — *  1-2  .=  9 
|    64  —  64-4-40  —  12'  —  28 

I     2l6—  144  +  6O  —  '12  —  I2O 

Gfc. 


WE  omit  trying  the  negative  divifors,  fince  there 
are  three  changes  in  the  figns  of  the  propofed  equa 
tion,  and  therefore  none  of  its  roots  can  be  negative  : 
And  fince  none  of  the  divifors  fucceed,  except  2  $ 
it  follows,  that  2  is  the  only  rational  root  of  the  equa 
tion,  the  other  two  being  either  irrational,  or  impof- 
fible. 

Let  it  be  required  to  find  the  roots  of  the  equa 
tion  x3  +  ix*  —  40*  +64  —  0. 

Here  the  divifors  of  the  laft  termy  .are  i,  2,  4,  8, 
16,32,  which  Jubflitutedjucceffi-vely  for  x  in  the  fro- 
fofed  equation, 

c  !  +  a  —  4o  +  64  =  27 

Gives,  <    84-8  —  80  +  64—0 

(,  64  +  32  —  160  4-  64  zr  o 

WHERE  the  only  divifors  that  fucceed,  are  2,  and 
14  ;  and  fince  there  are  but  two  changes  in  the  figns 
of  the  propofed  equation,  there  mud  be  one  negative 
root  :  We  are  therefore  to  fubftitute  the  divifois 
negatively  taken,  in  order  to  difcover  the  other  value 
of#;  and  on  trial,  we  find  that—  8  fucceeds.  There 
fore  the  three  roots  of  the  propofed  equation,  are+2 


BUT  when  one  of  the  roots  of  an  equation  is  found, 
the  relt  of  the  roots  may  be  found  with  lefs  trouble, 
by  dividing  the  propofed  equation  by  the  fimple 
equation,  deduced  from  the  root  already  found,  and 

finding 


(      37Q      ) 

rinding  the  roots  of  the  quotient,  which  will  be  an 
equation  a  degree  lower  than  the  propofed  one. 

THUS,  in  the  laft  example  the  root  -f  2  firft  found, 
gives  x  n:  2  ;  or.  # —  2  ~  o,  by  which  dividing  the 
propofed  equation  :    Thus, 
#  —  2)  x*  -h  ix*  —  40*  4-  64(.v*  +  4#  —  32. 


*  —  40,*? 

*  —  8* 


—  32*4-  64 

—  32*4-  64 


The  quotient  will  be  a  quadratic  equation  x*  -(- 
4#  —  32  ~  o  i  which  is  the  product  of  the  other  two 
funple  equations,  from  which  the  propofed  cubic  was 
generated  -,  and  whofe  two  roots  are  confequently, 
two  of  the  roots  of  that  cubic.  But  the  two  roots  of 
the  quadratic,  are  +4'an(l.— 8.  Therefore,  the 
three  roots  of  the  cubic  equation,  are  2,  4,  —  8,  the 
fame  as  before. 

THE  finding  all  the  divifors  of  the  laft  term  of  an 
equation,  efpecially  if  that  term  be  large,  is  much  fa 
cilitated  by  the  following 

R  U  L  £. 

t.  DIVIDE  the  laft  term  by  its  lead  diyifor  that  ex 
ceeds  .unity,  and  the  quotient  by  its  leaft  divifor ; 
proceeding  in  this  manner,  till  you  have  a  quotient 
that  is  not  farther  divifible  by  any  number  greater 
than  an, unit:  And  this  quotient  together  with  thole 
rs,,.  are.  the.  Srft  dwfprs  of  the  lafl  term. 

2, 


(     37 1      ) 

.  ; 

2.  FIND  all  the  produ&s  of  thofe  divifors  which 
arife  by  combining  them  two  and  two,  and  all  the 
produces  which  arife  by  combining  them  three  and 
three,  and  fo  on,  until  the  continued  product  of  the 
firft  divifors,  is  equal  to  the  quantity  to  be  divided  ; 
and  you  will  have  the  divifors  required. 

EXAMPLES. 

Thus,  fuppofe  the  laft  term  of  an  equation  to  be 
60:  Then  60-4-2  =  30,  30+2  zz:  15,  i$-r-3~:5; 
therefore,  2X2,  2X3>  2  X  5,  and  3  X  5,  are  the 
combinations  of  the  twos ;  and  2x^X3,  2X2X5? 
?  X  3  X  $,  tne  combinations  of  the  threes  ;  alfo, 

2X2X3X51  is  the  combination  of  the  fours  —  their 
continued  produ6t,  equal  to  the  quantity  to  be  di 
vided.  Therefore  all  the  divifors  of  60,  are  2,  3,  5, 

.4,  6,  10,  15,     12,  20,30,    60. 

And  in  like  manner,  the  divifors  of  i cab,  are  2,  5, 

ay  by  10,  20,  ib,  $a,  $b,  ab>  ioa>  $ab,  lab  and  ioab. 

BUT  there  is  another  method  for  the  reduction  of 

^equations  by  divifors,  which  is  lefs  prolix,  by  re 
ducing  the  divifors  to  more  narrow  limits,  by  the  fol 
lowing 

R  U  L  E. 

1.  INSTEAD  of  the  unknown  quantity  in  the  pro* 
'pofed  equation,    fubftitute  fuccefllvely  the  terms  of 
theprogreilion,  t,  o,  -^- 1,  &c.  and  find  all  the  divi- 

;  fors  of  die  fums  that  refult  by  fuch  fubftitution. 

2.  TAKE  out  all  the  arithmetical  progrefiions  that 
hca-n 'be  found   among  thofe   divifors,  whofe   terms 

correfjpdncl  v?idi  the  order  of  the  terms,  i,  o,  —  i. 


C     37*      ) 

&c.  and  common  difference  unity;  and  the  values 
of  A;  will  be  found  among  the  divifors  which  arife 
from  the  fubftirution  of  x  ~o,  that  belong  to  thofe 
progreffions. 

Note.  When  the  arithmetical  pregrejjion  is  increefing 
according  to  the  order  cf  the  terms  i,  o,  —  i,  the 
value  ofx  will  be  affirmative  -}  but  when  the  arith 
metical  progrejfion  is  decreajing,  the  'value  ofx  will 
be  negative. 

EXAMPLES. 

Let  x3  — x*  —  10  x  +  6  ~  o,  be  the  propofed 
equation  -,  and  by  fubftituting  fucceflively  for  x>  the 
terms  i,  o,  —  i,  the  work  will  Hand  as  follows. 

Suppojttions.  Refults.  Divifors.     Ar.P* 


r_4 
x  —  o      >#3— - #*  —  io#-r-6  —  <  4-6 

l+H 


HERE  the  progrefTion  is  decreafing,  and  3,  that 
term  which  (lands  againft  the  fuppofition  of  x  z:  o ; 
therefore,  —  3,  fubftituted  for  x  in  the  propofed 
equation,  gives,  —  27  —  9  +  30  +  6  —  o  ;  where 
all  the  terms  vanifhing,  it  follows,  that  —3  is  one 
of  the  roots  of  the  propofed  equation  5  and  2-f-x/2, 
and  2  —  v/  2,  the  other  two  roots,  found  by  dividing 
the  propofed  equation  by  #+3,  and  refolving  the 
quadratic  quotient. 

Suppofe  it  be  required  to  find  the  roots  of  the 
equation  v*  +  3?>3  —  19  «z;*  —  27^  +  90  —  0. 

Then  by  fubflituting  as  before^  the  work  will  Hand 
as  follows. 


V  ~  I 

<u  no 

v  i=  —  i 


Refvlt 
48 

90 
96 


(      373      ) 

Divifors. 

1.2,3,4,6,  &c 
1,2,  3,5,  6,  &c 


.  Progref. 


3,  5 


HERE  are  five  arithmetical  progrefTions ;  and  Tub- 
ilituting  2,  3,  —  3,  —  5,  refpeclively  for  i;  in  the 
propofed  equation,  the  whole  vanifhes  ;  the  other 
progreflion  being  in  this  cafe  ufelefs,  fince  the  num 
ber  of  roots  are  but  four.  Confcquently,2,  3,  — 3, 
—  5,  are  the  four  roots  required. 

THERE  are  many  other  methods  befide  thofe  which 
we  have  here  given  for  the  refolution  of  equations  $ 
which  the  confined  limits  of  our  plan  obliges  us  to 
omit,  and  proceed  to  dif cover  the  roots  of  equations 
by  the  method  of  approximation. 


CHAP.      XXL 

fbe  FINDING  the  ROOTS  of  NUMERAL 
E ^UAriO NS  in  GENE RAL,  ly  the  ME- 
?HOD  of  APPROXIMATION. 

ALTHOUGH  there  are  other  methods  for 
the  refolution  of  equations,  than  thofe  given  in 
the  iaft  chapter,  yet  the  moft  of  them  are  either  very 
prolcx,  or  confined  to  particular  cafes ;  but  the  fol 
lowing  method  of  approximation  is  general,  and  ex 
tends  to  numeral  equations  of  all  kinds  whatever, 
and  though  not  accurately  true,  gives  the  value  of 
the  root  to  any  afligned  degree  of  exactnefs  you 
pleafe,  by  the  following 

Bbb  RULE. 


374 


RULE. 

t.  FIND  by  trial,  a  number  nearly  equal  to  the 
root  required,  and  call  it  r  ;  and  put  x  for  the  dif 
ference  between  the  real  root  and  that  already  found, 
then  will  r  ±  x  •—  v. 

2.  INSTEAD  ot'v  and  its  powers  in  the  propofed 
equation,  fubftitute  r  ±  x  and  its  powers  j  and  there 
will  ariie  a  new  equation  involving  x  and  known 
quantities. 

3.  THEN  by  rejecting  all  the  terms  of  this  new  e- 
quation  that  involve  the  powers  of  A?;   and  aflum- 
ing  the  reft  equal  to  nothing,  the  value  of  A?  will  be 
determined  by  means  of  a  fimple  equation. 

4.  ADD  the  value  of  x  thus  found  to  r,  and  you 
•will  have  a  nearer  value  of  the  root  required  ;  which 
if  not  fufficiently  exact,  repeat  the  operation,  by  fub- 
ilituting  this  value  for  r  in  the  formula  exhibiting 
the  value  of  AT,  and  it  will  give  a  correction  of  the 
root;  which  if  not  yet  exact  enough,  proceed  to  a 
third  correction  -,  and  fo  on,  to  any  afligned  degree 
of  exadtnefs. 

EXAMPLES. 

Given,  *v*  4-  6v  —  31  ~  o,  to  find  v  by  approx 
imation. 

ne  root  found  by  trial  is  nearly  equal  to  3  : 
Therefore,  rn  3,  and  r  +  x  ~  v  : 


=  6r+6x 


6r  —  31  =  o  : 


(      375      ) 
Whence,  x—  •*  *  ~  r  ~  r—(ty  writing  $forr 

its  ^al)3l~^~l     =  ±  -  -3  i  and  v  =  3.3 

yf»</  if  3.3  ttjubjtitufedfar  r  in  the  equation,  x^-i. 
3i~^-6r    mJbM  bave  x  =  31-10.89-19:8 
2r  +  6  6.6  +  6 

—  .!£—  .  ~  .0246,  0r  rather  x  ir  .0245, 
12.6 


- 

Again,  if  this  value  bejubftitutedfor  r>  we  Jhall 
have  x  r:  .000005,  and  v  -=zr  -\-  x  ~3-3245°5>  for  a 
nearer  value  ofv  -,  and  Jo  on,  to  any  ajjtgned  degree  of 
exaflnefs. 

Given,  v3  +  2V  —  73  ^  o,  to  find  v  by  approx 
imation. 

<fhe  root  found  by  trial,  is  nearly  equal  4  : 
Therefore,  r  =  4,  and  r  -{-  z  n:  v  : 
?hen,vl  =:  r3  +  3^a2  +  3^2*  -i-s3- 


—73  -  —  73 


2r  +  22  —  73  rr  o  ;    $r, 

»7^  —  r3  —  <2r  73  —  64  —  8 

%  ,.•.,  -  ~  (by  writing  4  forz)  -  3  --  * 

3^+2  y    4^  +  2 

r=  —  =.02  s    ««^  therefore,    V  •=  r  +  z  —  4.02  ; 

^7«^  writing  this  value  for  r,  in  the  equation  z  = 
73  —  r3  —  2r  t 

y*  +  2 

—   64.964808    —    8.04 


, 

haw  z  = 


(•   376      ) 

—  .004808 


4.019905  nearly. 

And  after  this  manner  of  reafoning,  we  may  obtain 
theorems  for  approximating  to  the  roots  of  pure 
powers. 

Thus,  if  A  be  a  given  quantity  whofe  n  root  is  required^ 
r  the  neareft  lefs  root  in  the  integers,  and  v  the  difjer- 
ence  between  r  and  the  root  required  :  Then  will  rn  -\- 


2  2 

tj,      \  *) 

r        J<y3,    &V.   ~  A\    and  aJTuming    v  = 


A—  rn 


3 

ji 

.  cr>  more  nearly  >  taking  the  three  frjl  terms, 

A-  rn  • 


-1  nrn~l  +  n  X'lZlrB~2      A-r* 


A-rn  nr 


I 


» 


»-  -  1 


A-rn 

n  —  i  n   >  and  fy  writing 

~^r-  X  A-rn 


(      377      ) 

y  for  A — r  ,   we  have  v  —  - 


«•*•    *+*=.! 

2r 

—  (by  reduction) the   theorem  for 

n     n  — •  i 

2 

approximating  to  the  value 'of  vy  which  added  to  rt 
will  give  a  correction  of  the  root  ;  which  if  not  fuf- 
ficiendy  near  the  truth,  the  operation  muft  be  re 
peated,,  by  fjbftituting  the  new  r  in  the  equation 
exhibiting  the  value  of  v. 

Thus,  for  example,  iuppofe  the  cube  root  of  3  is 
required. 

Here  r  =  i ,  the  near  eft  lejs  root  in  the  integers >  and 
r  -f-  v  —  root  required. 

Therefore,  v  =     ./?_..   =  ^-2—  =  1  =  .4>^ 


r  +  v  zr  T  +  .4^:  i  .4,  w^/V^>  fubftitutedfor  r,  and  th& 

„  operation  repeated,   v  will  be  found  ~  .0397  ;    there- 

fore,  r  +  v  —  1.4  +  .0397  —   1-4397  n 

c/*3,  very  near. 


CHAP.     XXII. 

CONCERNING     UNLIMITED    PROB 
LEMS. 

HAVING  gone  through,   and  explained  the 
methods  ufed  in  arguing  limited  problems,  or 
fucn  as  admit  of  but  one  folution  ;  it  remains  there 
fore,  that  we  fhew  the  learner  how  to  reafon  about 

thofc 


(•     378      ) 

thofe  problems  which  are  unlimited,  or  admit  of  va 
rious  anfwers. 

IT  was  obferved  In  Chap,  xv,  of  this  Book,  that 
when  the  equations  expreffing  the  conditions  of  the 
queftion,  are  lefs  in  number  than  the  quantities 
fought,  the  queftion  is  unlimited,  or  capable  of  in 
numerable  anfwers ;  yet  .all  the  poflibie  anfwers  in 
whole  numbers,  are  for  die  moft  part  limited  to  a 
determinate  number. 

As  queftions  of  this  nature  admit  of  fome  varia 
tions  as  to  their  general  folution  ;  we  lhall  therefore 
confider  them  in  the  following  problems. 

PROBLEM      I. 

*• 

Tofind  the  values  of  v  and  y  in  whole  numbers,  in 
tbe  equation  av  jh  by  ±  c=z  o  -3  where  a>  b  and  f>  • 
are  given  quantities. 

RULE. 

1.  REDUCE  the  given  equation  to  its  lead  terms> 
by  dividing  it  by  its  greater!  common  divifor. 

2.  FIND  the  value  of  v  from  the  given  equation  ; 
and  reduce  the  refulting  expreffion,   by   expunging 
all  whole  numbers  from  it,  until   c  be  lefs  than  ay 
and  the  co-efficient  of y  becomes  unity. 

3.  ASSUME  this  laft  refult  equal  to  fome  known 
whole  number,  and  the  expreffion  reduced,  will  give 
the  value  of  y  in  known  terms  -9  from  which  the  val 
ue  of  v  may  be  determined  in  the  given  equation. 

Note.  If  after  tbe  given  equation  is  divided  by  its 
greatefl  common  divifor,  the  co-efficients  of  the 
unknown  quantities,  are  commensurable  to  each 
ether,  the  qusftion  is  impojfible. 

EXAM. 


(     379      5 


EXAMPLES. 

Given,  109  —  87 — 36  ~  o,  to  find  v  and y  in 
whole  numbers. 

Fir  ft,  51;  —  4y  —  1 8  =  o,  ^j  dividing  the  whole  by 
t;  or,  $v  —  4-y  —  18. 

Put  W N  for  any  whole  number: 

4-V 

•*•  T  -*-     •»   r-     »  7  /%* 

^c?  queftion : 


±iZ-        .   3  +  47  .     ,v  3  +  47 

"  —  3  -j  --  >  therefore,  • 


=  W  '  N,per  axiom  9.  Alfo>  ^-  =  /^  TV:  Confequsnt- 

,     $y     3  +  4y  __y—3l 

fy,     —  ---  -  —  --  c~~"^*     ^     3  ferawom  9; 

jy  —  o 

therefore,   —  -  —  rr  »  ;  and  for  the  leaft  value  of 


,  ajfume  n  —  O,  ^;/^/  we  fljall  havey  —  3  —  5^  zz  O  j 


. 
=  3,  rf»^r  v  n  -  —  •  n  6. 


Given,  26*1;  -f-  i8y  31140,  to  find  v  andj  in  whole 
aumbers. 


rfty   13^  -4-  97  =  70  ^y  dividing  the  whole  ly  2  : 


j 


?  —  ~  ny 
Therefore,  -  -  -  ~  W  N,  per  axiom  9  ;    #//>, 


5  + 
_ 


i  c  -f-  1  1  y 
X,  7.   But,    *  = 

O 

2  4-  i2y 
therefore,  -  -  —  TF  TV",  p  er  ax.  9. 


-^  „,  Ar 

whence,  —•  --  —  —  —  =:  /F7V; 

y  —  2 

9  :  ^»^/  —  —  =  n  ;  or,  y  z:  i  j»  ^-  2  ;  and 
\j 


a/fuming  n  zz  o,   w^  have  y  ir  2, 

o 

4- 

I  OWE  my  friend  a  moidore,  have  nothing  about 
me  but  crowns,  and  he  has  nothing  but  guineas  :  How 
mud  we  exchange  thefe  pieces  of  money,  fo  that  I 
may  acquit  myfelf  of  the  debt  ?  A  moidore  being 
valued  at  27  fhillings,  a  crown  at  5  Shillings,  and  a 
guinea  at  21  fhillings. 

Put  x  ~  number  of  crowns,  andy  the  number  o 
yeas  : 

5^  —  2iy  zz  27  by  the  queftion  : 


5  +  4X  +  ^-^  '  ^JequentJy,  1±2  _ 
2  4-jy 


zz  #  >  or,  2  +  y  zz  577  ;  ^»^  affuming  n~ 

27  - 

kavey  zz  3,  //&^  number  of  guineas,  and  x~  - 

zr  1  8,  /j&tf  number  of  crowns.  Therefore,  I  mujt  gros 
my  friend  18  crowns,  and  he  muft  give  me  three  gui 
neas* 

Given, 


Given,  4#-f*  17^:1:2900,  to  find  all  the  poflible 
values  of  x  and^y  in  whole  numbers. 

Firlt,y  -  29°°  ~-±f  =  W  N  ;  but 
17 


10  —  AX  40  —  1  6x 

.  8. 


iy 
ax.  9.    Alfo*  -  ^WN\  Confequently,   --  -j. 


—i  therefore,  - 


a/ummg  this  lafl  equation  =i  ff,  ^<?  gef  x  =:  17%  —  6, 
where  >  ifnbe  taken  =  i,  wejhallhavex  —  17  —  6  = 

2900  —  4* 
1  1  /0r  if/&^  fcj^f  ^^/^^  <?/  ^,  ^»  Jf  —  -  "  -  =z 

1  68  /?r  /^^  greateft  value  of  y  :    Andfince  6  -J-  oc^.  iy 

z:  «,  /j  ^  wi^/f  number  ;  //  isflafa,  that  n  -\-  1  /j  /^ 

/r^  augment  of  6  +  x-r-i'l   in   whole  numbers-,  and 

therefore,  x  =  17^  +  17  —  6,    tbefatnd  value  of  x  ; 

2QOO  —  —  A^ 

which  fubftituted  for  x  in  the  equation  y  ~  -  :  - 

will  give  the  fecond  value  ofy  :  Or,  by  adding  17  Juc-+ 
cejfively  to  the  values  ofx,  and  Jubtrafting  ^from  thofs 
0fy,  we  Jhall  have  all  the  fojJibU  valuss  of  x  and  y  in 
whole  numbers^  as  follows  :  viz.  x  —  1  1,  28,  45, 
to  708  j  andy  zz  168,  164,  160,  &fr.  /<?  4. 

PROBLEM       II- 


To  find  the  leafl  whole  number  AT,  /^»^/  being  divided 
by  the  given  numbers,  a>b,  c,  d>  &c.  Jhall  leave  given, 
remainders,  g,  k>  I,  m>  n,  G?Vt 

C  c  c  RULE, 


(      3**      ) 


RULE. 

1.  SUBTRACT  each  of  the  remainders  from  #,  and 
divide  the  feveral  refults  by  their  refpe&ive  divifors, 
ay  by  c,  d>  &c.  and  the  refuhing  quotients  will  equal 
whole  numbers. 

2.  ASSUME  the  firfl  equation  equal  b,  and  find  the 
value  of  x  in  terms  of  h. 

3.  SUBSTITUTE  the  value  of  x  in  terms  of£,  in  the 
fecond  equation  j  and  proceed  with  the  refult  as  in 
the  lad  problem,  by  expunging  all  whole  numbers, 
until  the  co-efficient  of  h  becomes  unity,  &c. 

4.  PUT  this  expreflion  equal  />,   and  find  the  value 
of*  in  terms  of/>,  by  means  of  the  equation  of  b. 

5.  SUBSTITUTE  the  value  of  x  in  terms  of/>,  in  the 
third  equation,  with  which  proceed  as  before,  and  fo 
on,  through  all  tht  given  equations  ;  affuming  the 
final  refult  equal  to  fome  known  whole  number,  and 
finding  the  values  of  the  feveral  fubflituted  letters,  £, 
/>,  &c.  from  which  the  value  of  A;  may  be.  determin 
ed  in  known  terms. 

EXAMPLES. 

To  find  the.  leaft  whole  number,  that  bemg  di 
vided  by  7  fhall  leave  6  remainder  j  but  being  divid 
ed  by  6  fhall  leave  4  remainder. 

Put  v  z:  number  Jought. 


i)  —  6 
djfvnu  -  ~  b)  and  we  Jh  all  have  v  —  yb  + 

which  Jubftifuttd  for  v  in  thejeccnd  equation,  gives 


(      38,1      ) 

6b 
But,  ~g~  WN:    Confequently, 

6h       £  +  2  b+z 

_-  _—  -  w  N)  md  ajfuming  ~ 

zr  77,  we  Jhall  have  h~  6n  —  2  ;  wfor*  ;/#  fo  /tfte 
=  i,  ^^  ^^77  ^i;<?  b  ~  4,  ^«df  «y  iz  7^  +'  6  z:  34, 
the  number  required. 

To  find  the  leaft  whole  number,  that  being  divid 
ed  by  1  8,  fhill  leave  14  remainder  -y  but  being  di 
vided  by  28,  fhall  leave  20  remainder. 

Put  v  ~  number  fought. 

v  —  14.  v  —  20 

Men,        ^      =PTN:  And,      ^      =:  W  N. 

v  —  \^ 
A[jumey          „  —  zi  b  -3  and  we  have  v~  18^4-14, 

which  fubftituted  for  v  in  the  fecond  equationy  gives 

—  6  ah  —  i 

—  -WN-,  or,  '    l        —WNby  dividing  all 

. 

the  terms  ly  2  ;  and 


—  -  X  2  =:  —  =  W  N.    Consequently  ,  ~  -- 

—  9       h  -f-  Q  ^  "4*  9 

;  -  —  »^  -  ==  ^  jy  .  ^^^  affuminv  -  ~  »> 
14  14  14 


w*  have  h  rr  i4»-—  9;  /?«^  putting  n  ~  i, 
^  zz  14^  —  9  —  5,  ^^^/  i;  ~  1  8^  4-  14= 
number  required. 

Diophantine    Problems. 

DIOPHANTINE  Problems,  fo  called,  from  Diophan* 
tus  their  inventor,  are  fuch  as  relate  to  the  finding  of 
Pquare  and  cube  numbers,  &c. 

THESS 


THESE  problems  are  fo  exceedingly  curious,  that 
Nothing  lefs  than  the  moft  refined  Algebia,  applied 
with  the  utmoft  fkill  and  judgment,  could  ever  fjr- 
mount  the  difficulties  which  neceffarily  attend  their 
folution.  The  peculiar  artince  made  ufe  of  in  form 
ing  fuch  pofitions  as  ihe  nature  of  the  problems  re 
quire,  fhews  the  great  ufe  of  Algebra,  or  the  analy 
tic  arr,  in  difcovering  thofe  things  that  otherwile, 
would  be  without  the  reach  of  human  uoderftandingo 

ALTHO  no  general  rule  can  be  given  for  the  foi-j- 
tipnofthefe  problems;  yet  the  following  direction 
will  be  very  ferviceable  on  many  occations. 

DIRECTION. 

ASSUME  one  or  more  letters,  for  the  root  of  the 
required  fquare,  cube,  &"c.  fuch  that  when  involved 
tu  t./e  height  of  the  propofed  power,  either  the  given 
nu nber,  or  the  highelt  term  of  the  unknown  quan 
tity  mav  vanifh.  Then  if  the  unknown  quantity  in 
the  reful tin.->  equation,  be  of  fimple  dimenfion,  find 
its  value  by  reducing  the  equation.  But  if  the  un 
known  quantity  be  ilill  a  fquare,  cube,  or  other  pow 
er  ;  affui-ne  or  her  letter  or  letters,  with  which  pro 
ceed  as  before,  until  the  higheft  term  of  the  unknown 
quantity  become  qf  fimple  dimenfion  in  the  equa 
tion. 

EXAMPLES. 

To  find  a  fquare  number  #%  fuch  that  A?*  +  i  lhall 
be  a  fquare  number. 

Affume  x —  2  for  the  root  ofxz-\-i  : 
Then  will  x — 2!*  n  x*  +  i  ;  that  is,  x*  —4*  +  4 
~  x'*  +  i  ;  or,  4*  n  4  — 1~3  i  whence ,x  r:^,  an& 


a  5     . 

w  • 

9 
TT 


is  tie  numler  required.    But  if  we  had  aJJumeX 

ar2  4-  i 

/<?r  A;'-,  wefhould  kavehad 
4r* 

1 


,  .  7    .         ... 
+  i  =—  -  1  --  *    ^wttB  /j  evidently  a  fquare 

number  ;  w£*r*  r  »*0y  be  taken  for  any  number. 

TO  6nd  two  nu/noers,  fuch  chat  chur  product  and 
quotient  may  be  both  fquare  and  cube  numbers* 

Ajjume  v9  and  v*  for  the  required  numbers  : 

Then  v9  x  ^3  —  'yl  *>  ^^  i>9-f-i;3  =^6,  ^r<?  m- 
J<?w//>  fquare  and  cube  numbers  ;  where  v  may  be  any 
number  taken  at  plea  fare. 

To  find  four  fquare  numbers  in  arithmetical  pro- 
greffion. 

For  the  fum  of  the  two  extremes,  ajjiime  in*  5  then 
will  thefum  of  the  two  means  be  aijo  2»a  by  the  nature 
of  the  proportion  : 

For  the  roots  of  the  two  means  9  ajfume  n  -f  3%,  and 
n  —  42  :  _  _ 

fben  will  n  -f  3zjz  +  n  —  42!'  =  2/2*  : 

That  is,  n*  +  6nz  4-  9^  -h  »x  -  *nx  +  i6z*=i 


-  2WZ  -4-    272*   =  2W*  t 

Or,  25  2  x  =  2»%  ;  and  by  dividing  by  z,  we  have  252 


Whence,  z~  in~i$  ;  andputingn—  i,  we  have 


,         6  lit 


And  for  the  roots  of  the  two  extremes,   ajfums  n 
n  4-  z  : 

Then  will  n~^-  . 
Or,  wz  — 

And  by  reduction,  z  '~  2»-j-j  ~  T; 


Whence,  n  -  -22)%  and  n  4-  zj*  zr  ~ 
'two  extremes.    So  that  the  four  fquare  numbers  in  arith 
metical  progrejfion,  are  ^T>  ^44,  |4|,  *|. 

To  find  a  number,  fuch  that  being  multiplied  with 
one  tenth  part  ofitfelf,  and  the  product  increafed  by 
36,  fhall  produce  a  fquare  number.  f 

jP«/  1;  for  the  number  fought  ;  /to  v*  -MO  +  36,  *V 

/<?  £<?  6i  fquare  number  : 

AJTuyne'the  root  of  this  fquare  ~  v  —  6,   then  will 

*v  "—  -'  (  |    rz:  ^"-f-io  -f-  36  ;  /^7/  /j,  'y1  —  i2i;-}-36i^ 
^^^-I0  «f  36  : 

Or,  1  01;  z  —  i  zov  zr  i;  2  j  tobence,  by  redutJion  v  n 
'•5-%  ^  number  required. 

To  divide  a  given  number  29,  confiding  of  two 
known  fquare  numbers  4  and  25,  into  two  other 
fquare  numbers. 

For  tbs  rcot  cf  the  frU  Jquarey  aJJ'ume  rv  —  25  and 
for  the  root  of  thefeccnd  nv  —  •  5  : 

will  rv  —^  4-  KV  —  J(a  n  29  : 


'v*  —  4^1;  4-4-4-  »*va  —  io»v  4-25  — 


Or,  r2  H-  «a<^*  —  4r—  IO/VT;  4-   29=  295    or, 
r*  4-  /22<y*  zr  4r4-  io»i;  ;  ^^  ^y   dividing  by    v,  we 
bave  r2-  4-  »2i;  zr  4^  4-  iotf  : 
4r  4.  1  6» 

Or,    *y  ir  —  TT~T"  s  .  ^^^  therefore,    rv  —  2  zz 


—  2  n  -  i  —  •  —  -a  -  :  and 
r*  4-  «a 

—  cra  4-  c#a 


^  n  2.  wejhall  have 


(      38?      ) 

14 

—  for  the  root  ofthefirfifquare,  and 

5  *      i  *» 

zi    y  /<?r  J&*  r<?0/  <?/  thefecond. 

To  find  three  iquare  numbers  in  arithmetical  pro* 
grefiion. 

AJfume  n*  for  the  mean  -,  then  will  in*  ~  thefum 
of  the  extremes  by  the  nature  of  the  proportion. 

For  the  root  of  the  greater  extreme,  ajjume  n  -j-  av, 
and  for  the  root  of  the  lefs  n  —  3-1; : 

Then  will  w— 3^]*-  +  »+  2-yl*  zz  2»* : 

FbattSy  n*  —  6»<z;  +  91;"  +  M*  +  4^»  -f  4^*  — 2«a: 

Or,  131;*  =  2»v  j  ^^^  4x  dividing  by  v,  we  have 
13^  n  2^  5  whence,  v  =  in~-\2>  where  n  may  be 
any  number  at  pleajure  : 

^fw^  4x  ajjuming  n  zi  i,  we  Jhall  have  v  zr  2-^-13  : 

dt*     40- 

Whence,    i  — —    zr— r1-  Tor  w  ^^/*  extreme  : 
*3|       109 


And  i  +  ~    —         >r  the  greater  :  Wherefore, 


49  289 

the  numbers  required^  are  -^*>   i, 


THE    END    OF    VOLUME  I. 


Os  •-    O-^\w 

OO     N  V 

t^.  i^,  O 

VO    ON  - 


;SEtP 

NO®     O     O    00     N  VO 


O    ^  r^.  O   '^^e 
r*"j  J-X.VO    —    O  '-^ 

oo  ee   N  'sO 
c«^  ^  w  -»f-  oo 


u^  o   P^  MI  -t 


«H    ro  r^OO    1-1-^-4- 
*•   O   t-^\  r^  r^.  O 

»•  oo  oo   o'o 
vo  *^-  ON 


-IN.  '.>    rn   -     rv.    *7s 

o  -4-  -*-  o   ^   ^- 


. 
O     O 


o 
a. 


vo 


~ 

D  vo  —  o  «">  o  o 
ON  r^vn  r^  N  •<*•  *f* 
r^~  f-o*o  ON  oo  r^vo 
NO  o  -4-  r^.  r-vo  — 

t-     O     O     N  NO     O     •«+• 


i/^so    O 
i«    t>x  ON 


•<*•  N     O 


VO    LO  N    ONOO   rn  rh  r^,  o   —  NO 

N    O    O    ry*>  "">  •*    i<^  f^  N  OO 

in    rj-'sO     "N  N  OO     ^t-  r^-0© 

«\CNO    ^-O\»^.O 
N    O    —    rv,  >-i 


•-  NO  NO 


O\ 


*»  ^ 


N     ThOOsO     N     rj-OONC3     M 

N  VA  •*  t*  ^r  ON  ON 


-     N     *i-  00  NO 


....  .-...- 

^; 


C\ 


